ICS 6B Midterm 1

Laws of Propositional Logic: Associative Laws

( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r )

What is the order of operations in absence of parentheses for compound propositions?

1. ¬, ∀, and ∃ 2. ∧ 3. ∨ 4. → and ↔

Contradiction

A compound proposition is a contradiction if the proposition is always false, regardless of the truth value of the individual propositions that occur in it. The proposition p ∧ ¬p is an example of a simple contradiction, because the proposition is false regardless of whether p is true or false. Showing that a compound proposition is not a contradiction only requires showing a particular set of truth values for its individual propositions that cause the compound proposition to evaluate to true.

Tautology

A compound proposition is a tautology if the proposition is always true, regardless of the truth value of the individual propositions that occur in it. The proposition p ∨ ¬p is a simple example of a tautology since the proposition is always true whether p is true or false. Showing that a compound proposition is not a tautology only requires showing a particular set of truth values for its individual propositions that cause the compound proposition to evaluate to false.

Compound Proposition

A compound proposition is created by connecting individual propositions with logical operations. A logical operation combines propositions using a particular composition rule.

Conditional Proposition

A compound proposition that uses a conditional operation is called a conditional proposition. A conditional proposition expressed in English is sometimes referred to as a conditional statement, as in "If there is a traffic jam today, then I will be late for work." In p → q, the proposition p is called the hypothesis, and the proposition q is called the conclusion.

Counterexamples For Conditional Statements

A counterexample for a conditional statement must satisfy all the hypotheses and contradict the conclusion. Using the language of logic, consider the expression that says for every element x in a particular set (or domain), if the hypothesis H(x) is true for x, then the conclusion C(x) must also be true for x: ∀x (H(x) → C(x)) A counterexample for the expression above is a specific element d in the domain for variable x such that (H(d) → C(d)) is false. Using the laws of logic, it can be shown that (H(d) → C(d)) is false if and only if H(d) is true and C(d) is false. Therefore a counterexample is a particular element of the domain that satisfies all the hypothesis of a conditional statement and does not satisfy the conclusion. A counterexample for a statement with more than one hypothesis, such as ∀x ((H1(x) ∧ H2(x)) → C(x)), must be a particular element d that satisfies all the hypotheses and does not satisfy the conclusion: H1(d) and H2(d) are both true and C(d) is false.

Quantified Statement

A logical statement that includes a universal or existential quantifier is called a quantified statement.

Predicate

A logical statement whose truth value is a function of one or more variables is called a predicate. The statement "x is an odd number" is not a proposition because the statement does not have a well-defined truth value until the value of x is specified. If P(x) is defined to be the statement "x is an odd number", then P(5) corresponds to the statement "5 is an odd number". P(5) is a proposition because it has a well defined truth value. The domain of a variable in a predicate is the set of all possible values for the variable. For example, a natural domain for the variable x in the predicate "x is an odd number" would be the set of all integers. If the domain of a variable in a predicate is not clear from context, the domain should be given as part of the definition of the predicate. If all the variables in a predicate are assigned specific values from their domains, then the predicate becomes a proposition with a well defined truth value. Another way to turn a predicate into a proposition is to use a quantifier.

Particular Element

A particular element of the domain may have properties that are not shared by all the elements of the domain. For example, if the domain is the set of all integers, 3 is a particular element of the domain. The number 3 is odd which is not a property that is shared by all integers.

Proofs, Theorems, and Axioms

A primary endeavor in mathematics is to prove theorems. A theorem is a statement that can be proven to be true. A proof consists of a series of steps, each of which follows logically from assumptions, or from previously proven statements, whose final step should result in the statement of the theorem being proven. The proof of a theorem may make use of axioms, which are statements assumed to be true. A proof may also make use of previously proven theorems. Although mathematical proofs are typically expressed in English, the formalism of logic provides a good foundation for mathematical reasoning used in proving theorems. A proof should read like a verbal argument designed to convince a skeptical listener that an assertion is true. Although mathematical theorems are usually expressed in English, translating a statement into logic is often helpful in understanding exactly what the statement means. Understanding whether a theorem to be proven is a universal or existential statement is an important first step in proving that the theorem is true.

Proof By Contradiction

A proof by contradiction starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of the assumption. The reasoning behind proof by contradiction is that if the assumption that the theorem is false leads to a conclusion which cannot be true, then the theorem must be true. A proof by contradiction is sometimes called an indirect proof. If t is the statement of the theorem, the proof begins with the assumption ¬t and leads to a conclusion r ∧ ¬r, for some proposition r. A proof by contradiction often starts with: "Suppose ¬t", where ¬t is a statement that is equivalent to the negation of the theorem. A proof by contradiction can be used to prove theorems that are not of the form p → q.

Proof By Contrapositive

A proof by contrapositive proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true. In other words, ¬c is assumed to be true and ¬p is proven as a result of ¬c. Many theorems are conditional statements that also have a universal quantifier such as: For every integer n, if n^2 is odd then n is odd. The domain of variable n is the set of all integers. If D(n) is the predicate that says that n is odd, then the statement is equivalent to the logical expression: ∀n (D(n^2) → D(n)). A proof by contrapositive of the theorem starts with n, an arbitrary integer, assumes that D(n) is false, and then proves that D(n^2) is false. Some conditional statements have more than one hypothesis. In a proof by contrapositive it is only necessary to show that one of the hypotheses is false, assuming that the rest of the hypotheses are true and the conclusion is false. For example, consider the conditional statement: If H1 and H2 are both true then C is true. [(H1 ∧ H2) → C] The contrapositive of this conditional statement is: If C is false, then it cannot be the case that H1 and H2 are both true. [¬C → ¬(H1 ∧ H2)] Sometimes using the following forms of the statement can make it easier to write the proof. Assume that C is false and H1 is true. We shall show that H2 is false. It is also valid to swap the roles of H1 and H2 and start the proof with: Assume that C is false and H2 is true. We shall show that H1 is false. Remember, it is only necessary to show that one of the hypotheses is false.

Proving Existential Statements

A proof that shows that an existential statement is true is called an existence proof. The most common type of existence proof is a constructive proof of existence. An existential statement asserts that there is at least one element in a domain that has some particular properties. A constructive proof of existence gives a specific example of an element in the domain or a set of directions to construct an element in the domain that has the required properties. A nonconstructive proof of existence proves that an element with the required properties exists without giving a specific example. A common method for giving a nonconstructive existence proof is to show that the non-existence of an element with the required properties leads to a contradiction.

Proposition

A proposition is a statement that is either true or false.

What are the number of rows needed in a compound proposition's truth table?

A truth table for a compound proposition has a row for every possible combination of truth assignments for the statement's variables. If a compound proposition has n variables, there are 2^n rows. Ex: The truth table for compound proposition (p ∨ r) ∧ ¬q has 2^3 = 8 rows. Technique to fill in the truth table: To fill in the variable columns, each column is filled in from top to bottom, beginning with T. Start with the right-most variable column and fill in the squares with an alternating T and F pattern. The next column to the left is filled in by an alternating TT and FF pattern. The next column to the left is filled in by an alternating TTTT and FFFF pattern. For each new column, the number of T's and F's in the pattern is doubled.

Truth Table

A truth table shows the truth value of a compound proposition for every possible combination of truth values for the variables contained in the compound proposition. Every row in the truth table shows a particular truth value for each variable, along with the compound proposition's corresponding truth value.

Free vs Bound variables

A variable x in the predicate P(x) is called a free variable because the variable is free to take on any value in the domain. The variable x in the statement ∀x P(x) is a bound variable because the variable is bound to a quantifier. A statement with no free variables is a proposition because the statement's truth value can be determined. In the statement (∀x P(x)) ∧ Q(x), the variable x in P(x) is bound by the universal quantifier, but the variable x in Q(x) is not bound by the universal quantifier. Therefore the statement (∀x P(x)) ∧ Q(x) is not a proposition. In contrast, the universal quantifier in the statement ∀x (P(x) ∧ Q(x)) binds both occurrences of the variable x. Therefore ∀x (P(x) ∧ Q(x)) is a proposition.

Arbitrary Element

An "arbitrary element" means nothing is assumed about the element other than the fact that it is in the domain. An arbitrary element of a domain has no special properties other than those shared by all the elements of the domain

Disproving Existential Statements

An existential statement asserts that there is at least one element in a domain that has some particular properties. In order to show that an existential statement is false, it is necessary to argue that every single element of the domain does not have the required properties. The approach to proving that an existential statement is false is the same as the approach to proving that a universal statement is true.

Using logic to express "everyone else"

Consider a scenario where the domain is a group of people who are all working on a joint project. Define the predicate M(x, y) that indicates whether x sent an email to y. The statement ∀x ∀y M(x, y) asserts that every person sent an email to every other person and every person sent an email to himself or herself. How could we use logic to express that everyone sent an email to everyone else without including the case that everyone sent an email to himself or herself? The idea is to use the conditional operation: (x ≠ y) → M(x, y). Given that M(Fred, Fred) and M(Marge, Marge) are both false (while every other pair for M is true), the statement ∀x ∀y M(x, y) is false. However the statement ∀x ∀y ((x ≠ y) → M(x, y)) is true, since the pair (Fred, Fred) leads to a conditional statement that is: F → F, which evaluates to T. The statement says that for every pair, x and y, if x and y are different people then x sent an email to y. That is, everyone sent an email to everyone else. For this statement to be true, the following has to be true: for every pair such that x ≠ y, M(x, y) is true.

Alternating Nested Quantifiers

Given the predicate M: M(x, y): x sent an email to y A quantified expression can contain both types of quantifiers as in: ∃x ∀y M(x, y). The quantifiers are applied from left to right, so the statement ∃x ∀y M(x, y) translates into English as: ∃x ∀y M(x, y) ↔ "There is a person who sent an email to everyone." Switching the quantifiers changes the meaning of the proposition: ∀x ∃y M(x, y) ↔ "Every person sent an email to someone." For the proposition: ∀x ∀y M(x, y), the proposition can be expressed in English as: ∀x ∀y M(x, y) ↔ "Everyone sent an email to everyone." For the proposition: ∃x ∃y M(x, y), the proposition can be expressed in English as: ∃x ∃y M(x, y) ↔ "There is a person who sent an email to someone."

Nested Quantifiers

If a predicate has more than one variable, each variable must be bound by a separate quantifier. A logical expression with more than one quantifier that binds different variables in the same predicate is said to have nested quantifiers.

Proofs of Universal Statements: Universal Generalization

If the domain of a universal statement is a large or even infinite set, it becomes impractical or infeasible to prove the statement individually for each element in the domain. For this reason, the most common method for proving universal statements is to use universal generalization. A proof that uses universal generalization to prove a universal statement names an arbitrary object in the domain and proves the statement for that object. "Arbitrary" means that nothing is assumed about the object other than the assumptions that are given in the statement of the theorem. The only way to be certain that a universal statement is true is a general proof that holds for all objects in the domain. A mathematician who does not know whether an unproven statement is true or false may divide his or her time between looking for a counterexample showing that the statement is false or a proof showing that the statement is true.

Proof By Exhaustion

If the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually. A proof of this kind is called a proof by exhaustion.

Rules of Inference with quantifiers

In order to apply the rules of inference to quantified expressions, such as ∀x ¬(P(x) ∧ Q(x)), we need to remove the quantifier by plugging in a value from the domain to replace the variable x. A value that can be plugged in for variable x is called an element of the domain of x. For example, if the domain is the set of all integers, ¬(P(3) ∧ Q(3)) is a proposition to which De Morgan's law can be applied. Elements of the domain can be defined in a hypothesis of an argument.

Two Player Game

In reasoning whether a quantified statement is true or false, it is useful to think of the statement as a two player game in which two players compete to set the statement's truth value. One of the players is the "existential player" and the other player is the "universal player". The variables are set from left to right in the expression. The Existential Player selects values for existentially bound variables and tries to make the expression true. The Universal Player selects values for universally bound variables and tries to make the expression false. If the predicate is true after all the variables are set, then the quantified statement is true. If the predicate is false after all the variables are set, then the quantified statement is false. Consider as an example the following quantified statement in which the domain is the set of all integers: ∀x ∃y (x + y = 0) The universal player first selects the value of x. Regardless of which value the universal player selects for x, the existential player can react to the universal player's selection by selecting y to be -x, which will cause the sum x + y to be 0. Because the existential player can always succeed in causing the predicate to be true, the statement ∀x ∃y (x + y = 0) is true. Switching the order of the quantifiers gives the following statement: ∃x ∀y (x + y = 0) Now, the existential player goes first and selects a value for x. Regardless of the value chosen for x, the universal player can react to the existential player's selection by selecting some value for y that causes the predicate to be false. For example, if x is an integer, then y = -x + 1 is also an integer and x + y = 1 ≠ 0. Thus, the universal player can always win and the statement ∃x ∀y (x + y = 0) is false.

Best Practices in Writing Proofs

Indicate when the proof starts and ends Write proofs in complete sentences: A proof should read like English text. In mathematical proofs, English sentences often contain mathematical expressions but those should read naturally as part of the sentence. For example: "If x is an integer that is greater than 0, then x ≥ 1." Give the reader a roadmap of what has been shown, what is assumed, and where the proof is going: The beginning of a proof should always state what facts are assumed. It can also be helpful to inform the reader what will be proven in the proof. If a proof is long, it is helpful to indicate at one or more points in the middle what has been proven and what has yet to be proven. For example: "We have shown that n is a positive integer. Now we must establish that n is composite." Introduce each variable when the variable is used for the first time: For example: "Since we know that m divides n, there is an integer k such that n = km." This sentence introduces the variable k. Variables m and n should already have been introduced. A block of equations should be introduced with English text and each step that does not follow from algebra should be justified: If the justification for a step does not fit easily on the line of the equation, the justification can be provided right after the block of equations.

Moving Quantifiers in logical statements

M(x, y): x is married to y. A(x): x is an adult. "For every person x, if x is an adult, then there is a person y to whom x is married." ∀x (A(x) → ∃y M(x, y)) Since y does not appear in the predicate A(x), "∃y" can be moved to the left so that it appears just after the ∀x resulting in the following equivalent expression: ∀x ∃y(A(x) → M(x, y)) Note a quantifier can not be moved in front of another quantifier without changing the meaning of the expression. For example, ∀x ∃y(A(x) → M(x, y)) is not logically equivalent to ∃y ∀x(A(x) → M(x, y)).

Direct Proofs

Many mathematical theorems take the form of a conditional statement in which a conclusion follows from a set of hypotheses. Theorems of this kind can be expressed as p → c, where p is a proposition which asserts that a set of hypotheses are true and c is the conclusion. Sometimes p is referred to as "the hypothesis" for simplicity. In a direct proof of a conditional statement, the hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption. Many theorems are conditional statements that also have a universal quantifier such as: For every integer n, if n is odd then n2 is odd. The domain of variable n is the set of all integers. If D(n) is the predicate that says that n is odd, then the statement is equivalent to the logical expression: ∀n (D(n) → D(n^2)). A direct proof of the theorem starts with n, an arbitrary integer, assumes that n is odd, and then proves that n^2 is odd.

Proof By Cases

Many theorems can be phrased as ∀x P(x), where the value of variable x can be any element from some domain. Sometimes proving such a theorem simultaneously for all elements in the domain is difficult, but the proof becomes more approachable if the domain is broken down into different classes where each class can be addressed separately. A proof by cases of a universal statement such as ∀x P(x) breaks the domain for the variable x into different classes and gives a different proof for each class. The proof for each class is called a case. Every value in the domain must be included in at least one class. In a proof by cases, the cases are numbered, and each case begins with "Case n:", where n is the number of that case. The number is followed by a statement of the assumptions for that case. Without loss of generality: Sometimes the proofs for two different cases are so similar, that it is repetitive to include both cases. When this happens, the two cases can be merged into one case. The term without loss of generality (sometimes abbreviated WLOG or w.l.o.g.) is used in mathematical proofs to narrow the scope of a proof to one special case in situations when the proof can be easily adapted to apply to the general case. Consider a proof of the following theorem: Theorem: For any two integers x and y, if x is even or y is even, then xy is even. A proof of the theorem could have two cases: one case assumes that x is even and the other case assumes that y is even. Since the assumption of the theorem is that at least one of x or y is even, the two cases cover all the possibilities for x and y. The proofs for the two cases would be identical except that the roles of x and y would be reversed. Instead a proof could address only one case and use the term "without loss of generality".

Universal Instantiation

Replaces a quantified variable with an element of the domain. Note that the rules of instantiation and generalization only apply to non-nested quantifiers. c is an element (arbitrary or particular) ∀x P(x) ------------------------------------------- ∴ P(c) Example: Sam is a student in the class. Every student in the class completed the assignment. ------------------------------------------------------------ Therefore, Sam completed his assignment.

Existential Instantiation

Replaces a quantified variable with an element of the domain. Note that the rules of instantiation and generalization only apply to non-nested quantifiers. ∃x P(x) ------------------------------------- ∴ (c is a particular element) ∧ P(c) Note: Each use of existential instantiation must define a new element with its own name (e.g., "c" or "d"). It is important to define a new particular element with a new name for each use of existential instantiation within the same logical proof in order to avoid a faulty proof that an invalid argument is valid. Example: There is an integer that is equal to its square. --------------------------------------------------- Therefore, c^2 = c, for some integer c.

Universal Generalization

Replaces an element of the domain with a quantified variable. Note that the rules of instantiation and generalization only apply to non-nested quantifiers. c is an arbitrary element P(c) ---------------------------- ∴ ∀x P(x) Example: Let c be an arbitrary integer. c ≤ c^2 -------------------------------------------------------------- Therefore, every integer is less than or equal to its square.

Existential Generalization

Replaces an element of the domain with a quantified variable. Note that the rules of instantiation and generalization only apply to non-nested quantifiers. c is an element (arbitrary or particular) P(c) ------------------------------------------- ∴ ∃x P(x) Example: Sam is a particular student in the class. Sam completed the assignment. ----------------------------------------------------------------- Therefore, there is a student in the class who completed the assignment.

Contrapositive Conditional Statements

The contrapositive of p → q is ¬q → ¬p Original Proposition: If it is raining today, the game will be cancelled. Contrapositive Proposition: If the game is not cancelled, then it is not raining today.

Converse Conditional Statements

The converse of p → q is q → p Original Proposition: If it is raining today, the game will be cancelled. Converse Proposition: If the game is cancelled, it is raining today.

The form of an argument

The hypotheses and conclusion in a logical argument can also be expressed in English, as in: It is raining today. If it is raining today, I will not ride my bike to school. --------------------------------------------------------- ∴ I will not ride my bike to school. The form of an argument expressed in English is obtained by replacing each individual proposition with a variable. While it is common to express a logical argument in English, the validity of an argument is established by analyzing its form. Define propositional variables p and q to be: p: It is raining today. q: I will not ride my bike to school. The argument can then be expressed back in logic: p p → q -------- ∴ q When arguments are expressed in English, the propositions sometimes have known truth values (such as whether a number like 7 is even or odd). Even if all of the hypotheses are true and the conclusion is true, the argument can still be invalid due to it's form.

Inverse Conditional Statements

The inverse of p → q is ¬p → ¬q Original Proposition: If it is raining today, the game will be cancelled. Inverse Proposition: If it is not raining today, the game will not be cancelled.

Argument

The language of logic allows us to formally establish the truth of logical statements, assuming that a set of hypotheses is true. An argument is a sequence of propositions, called hypotheses, followed by a final proposition, called the conclusion. Whenever the hypotheses of an argument are all true, the argument is valid if the conclusion is true and invalid if the conclusion is false. An argument will be denoted as: p1 p2 ... pn ---- ∴ c p1 ... pn are the hypotheses and c is the conclusion. The symbol ∴ reads "therefore." The argument is valid whenever the proposition (p1 ∧ p2 ∧ ... ∧ pn) → c is a tautology. In order to use a truth table to establish the validity of an argument, a truth table is constructed for all the hypotheses and the conclusion. Each row in which all the hypotheses are true is examined. If the conclusion is true in each of the examined rows, then the argument is valid. If there is any row in which all the hypotheses are true but the conclusion is false, then the argument is invalid (An argument can be shown to be invalid by showing an assignment of truth values to its variables that makes all the hypotheses true and the conclusion false). The final proof of invalidity only requires a single truth assignment for which all the hypotheses are true and the conclusion is false. According to the commutative law, reordering the hypotheses does not change whether an argument is valid or not. Therefore two arguments are considered to be the same even if the hypotheses appear in a different order.

Negating a Universal Quantifier

The negation operation can be applied to a quantified statement, such as ¬∀x F(x). If the domain for the variable x is the set of all birds and the predicate F(x) is "x can fly", then: "Not every bird can fly." ≡ "There exists a bird that cannot fly." ¬∀x F(x) ≡ ∃x ¬F(x), The equivalence of the previous two statements is an example of De Morgan's law for quantified statements, which is formally stated as ¬∀x F(x) ≡ ∃x ¬F(x).

Negating an Existential Quantifier

The negation operation can be applied to a quantified statement, such as ¬∃x A(x). If the domain for the variable x is the set of children enrolled in a class and the predicate A(x) is "x is absent today", then: "It is not true that there is a child in the class who is absent today." ≡ "Every child enrolled in the class is not absent today." ¬∃x P(x) ≡ ∀x ¬P(x) The logical equivalence of the previous two statements is an example of De Morgan's laws for quantified statements, which is formally stated as ¬∃x P(x) ≡ ∀x ¬P(x).

Allowed Assumptions in Proofs

The rules of algebra. For example if x, y, and z are real numbers and x = y, then x+z = y+z. The set of integers is closed under addition, multiplication, and subtraction. In other words, sums, products, and differences of integers are also integers. Every integer is either even or odd. If x is an integer, there is no integer between x and x+1. In particular, there is no integer between 0 and 1. The relative order of any two real numbers. For example 1/2 < 1 or 4.2 ≥ 3.7. The square of any real number is greater than or equal to 0.

Logical Proof

The validity of an argument can be established by applying the rules of inference and laws of propositional logic in a logical proof. A logical proof of an argument is a sequence of steps, each of which consists of a proposition and a justification. If the proposition in a step is a hypothesis, the justification is "Hypothesis". Otherwise, the proposition must follow from previous steps by applying one law of logic or rule of inference. The justification indicates which rule or law is used and the previous steps to which it is applied. For example, having justification "Modus ponens, 2, 3" on line 4 means that the Modus ponens rule is applied to the propositions on lines 2 and 3, resulting in the proposition on line 4. The proposition in the last step in the proof must be the conclusion of the argument being proven. In order to apply the rules of inference to quantified expressions, such as ∀x ¬(P(x) ∧ Q(x)), we need to remove the quantifier by plugging in a value from the domain to replace the variable x. A value that can be plugged in for variable x is called an element of the domain of x. For example, if the domain is the set of all integers, ¬(P(3) ∧ Q(3)) is a proposition to which De Morgan's law can be applied. Elements of the domain can be defined in a hypothesis of an argument. Elements of the domain can also be introduced within a proof in which case they are given generic names such as "c" or "d". Every domain element referenced in a proof must be defined on a separate line of the proof. If the element is defined in a hypothesis, it is always a particular element and the definition of that element in the proof is labeled "Hypothesis". If an element is introduced for the first time in the proof, the definition is labeled "Element definition" and must specify whether the element is arbitrary or particular.

The Language of Proofs

Thus/therefore/then/hence/it follows that: A statement that follows from the previous statement or previous few statements can be started with "Thus" or "Therefore". Other words that serve the same purpose are "it follows that", "then", "hence". Let: New variable names are often introduced with the word "let". For example, "Let x be a positive integer". Suppose: The word "suppose" can also be used to introduce a new variable. For example: "Suppose that x is a positive integer". Suppose is also used to introduce a new assumption, as in: "Suppose that x is odd", assuming that x has already been introduced as an integer earlier in the proof. Since/because we know that: If a statement depends on a fact that appeared earlier in the proof or in the assumptions of the theorem, it can be helpful to remind the reader of that fact before the statement. The phrase "because we know that" can serve the same purpose. By definition: A fact that is known because of a definition, can be started with the phrase "By definition". For example: "The integer m is even. By definition, m = 2k for some integer k." By assumption: A fact that is known because of an assumption, can be started with the phrase "By assumption". For example: "By assumption, x is positive. Therefore x > 0." In other words: Sometimes it is useful to rephrase a statement in a more specific way. The phrase "in other words" is useful in this context. For example: "We must show that the average of x and y is positive. In other words, we must show that (x+y)/2 > 0." Gives/yields: Sometimes a proof is clearer if even an algebraic step is justified. The words "gives" and "yields" are useful to say that one equation or inequality follows from another. For example: "Multiplying both sides of the inequality x > y by 2 gives 2x > 2y."

Logical Equivalence

Two compound propositions are said to be logically equivalent if they have the same truth value regardless of the truth values of their individual propositions. If s and r are two compound propositions, the notation s ≡ r is used to indicate that r and s are logically equivalent.

Rules of Inference known to be valid arguments: Addition

p -------- ∴ p ∨ q

Rules of Inference known to be valid arguments: Modus Ponens

p p → q -------- ∴ q

Rules of Inference known to be valid arguments: Conjunction

p q -------- ∴ p ∧ q

Rules of Inference known to be valid arguments: Hypothetical Syllogism

p → q q → r -------- ∴ p → r

Laws of Propositional Logic: Conditional Identities

p → q ≡ ¬p ∨ q p ↔ q ≡ ( p → q ) ∧ ( q → p )

Laws of Propositional Logic: Domination Laws

p ∧ F ≡ F p ∨ T ≡ T

Rules of Inference known to be valid arguments: Simplification

p ∧ q -------- ∴ p

Laws of Propositional Logic: Complement Laws

p ∧ ¬p ≡ F p ∨ ¬p ≡ T ¬T ≡ F ¬F ≡ T

Laws of Propositional Logic: Distributive Laws

p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r )

Laws of Propositional Logic: Absorption Laws

p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p

Laws of Propositional Logic: Identity Laws

p ∨ F ≡ p p ∧ T ≡ p

Laws of Propositional Logic: Idempotent Laws

p ∨ p ≡ p p ∧ p ≡ p

Rules of Inference known to be valid arguments: Disjunctive Syllogism

p ∨ q ¬p -------- ∴ q

Rules of Inference known to be valid arguments: Resolution

p ∨ q ¬p ∨ r -------- ∴ q ∨ r

Laws of Propositional Logic: Commutative Laws

p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p

Negation Operator (NOT)

¬ The negation operation acts on just one proposition and has the effect of reversing the truth value of the proposition. The negation of proposition p is denoted ¬p and is read as "not p".

Laws of Propositional Logic: De Morgan's Laws

¬( p ∨ q ) ≡ ¬p ∧ ¬q ¬( p ∧ q ) ≡ ¬p ∨ ¬q

Rules of Inference known to be valid arguments: Modus Tollens

¬q p → q -------- ∴ ¬p

Laws of Propositional Logic: Double Negation Laws

¬¬p ≡ p

Conditional Operator (THEN)

→ The conditional operation is denoted with the symbol →. The proposition p → q is read "if p then q". The proposition p → q is false if p is true and q is false; otherwise, p → q is true. A compound proposition that uses a conditional operation is called a conditional proposition. A conditional proposition expressed in English is sometimes referred to as a conditional statement, as in "If there is a traffic jam today, then I will be late for work." In p → q, the proposition p is called the hypothesis, and the proposition q is called the conclusion.

Biconditional Operator (IFF)

↔ If p and q are propositions, the proposition "p if and only if q" is expressed with the biconditional operation and is denoted p ↔ q. The proposition p ↔ q is true when p and q have the same truth value and is false when p and q have different truth values. Alternative ways of expressing p ↔ q in English include "p is necessary and sufficient for q" or "if p then q, and conversely". The term iff is an abbreviation of the expression "if and only if", as in "p iff q".

Universal Quantifier

∀ The logical statement ∀x P(x) is read "for all x, P(x)" or "for every x, P(x)". The statement ∀x P(x) asserts that P(x) is true for every possible value for x in its domain. The symbol ∀ is a universal quantifier and the statement ∀x P(x) is called a universally quantified statement. ∀x P(x) is a proposition because it is either true or false. ∀x P(x) is true if and only if P(n) is true for every n in the domain of variable x. If the domain is a finite set of elements {a1, a2, ..., ak}, then: ∀x P(x) ≡ P(a1) ∧ P(a2) ∧ ... ∧ ... P(ak) If the domain is the set of students in a class and the predicate A(x) means that student x completed the assignment, then the proposition ∀x A(x) means: "Every student completed the assignment." Establishing that ∀x A(x) is true requires showing that each and every student in the class did in fact complete the assignment. Some universally quantified statements can be shown to be true by showing that the predicate holds for an arbitrary element from the domain. Establishing that ∀x A(x) is false requires providing one counterexample. A counterexample for a universally quantified statement is an element in the domain for which the predicate is false. A single counterexample is sufficient to show that a universally quantified statement is false.

Existential Quantifier

∃ The logical statement ∃x P(x) is read "There exists an x, such that P(x)". The statement ∃x P(x) asserts that P(x) is true for at least one possible value for x in its domain. The symbol ∃ is an existential quantifier and the statement ∃x P(x) is called a existentially quantified statement. ∃x P(x) is a proposition because it is either true or false. ∃x P(x) is true if and only if P(n) is true for at least one value n in the domain of variable x. If the domain is a finite set of elements {a1, a2, ..., ak}, then: ∃x P(x) ≡ P(a1) ∨ P(a2) ∨ ... ∨ ... P(ak) If the domain is the set of students in a class and the predicate A(x) means that student x completed the assignment, then ∃x A(x) is the statement: "There is a student who completed the assignment." Establishing that ∃x A(x) is true only requires finding one particular student who completed the assignment. An example for an existentially quantified statement is an element in the domain for which the predicate is true. Establishing that ∃x A(x) is false requires showing that every student in the class did not complete the assignment. Some existentially quantified statements can be shown to be false by showing that the predicate is false for an arbitrary element from the domain.

Conjunction Operator (AND)

∧ The proposition p ∧ q is read "p and q" and is called the conjunction of p and q. p ∧ q is true if both p is true and q is true. p ∧ q is false if p is false, q is false, or both are false.

Inclusive Disjunction Operator (OR)

∨ The proposition p ∨ q is read "p or q", and is called the disjunction of p and q. p ∨ q is true if either one of p or q is true, or if both are true. The proposition p ∨ q is false only if both p and q are false.

Exclusive Disjunction Operator (XOR)

⊕ The exclusive or of p and q evaluates to true when p is true and q is false or when q is true and p is false.