Discrete Mathematics Exam 1 Review (Chapter 2 to Chapter 3)

Theorem: 3.2.2 Negation of an Existential Statement

"E' x in D, Q(x)" -> "V- x in D such that ~Q(x)" or equivalently, ~(E' x in D, Q(x)) _= V- x in D such that ~Q(x).

Generic examples for Interpreting Statements with Two Different Quantifiers

"Every nonzero real number has a reciprocal" -> ∀ nonzero real number x, ∃ real number y such that xy = 1. "There is a real number with no reciprocal" -> ∃ real number x such that for ∀ real number y, xy doesn't equal 1.

Definition: Only If

"If not q then p"

Definition: Sufficient Condition

"If r then s"

Definition: Necessary Condition

"If s then r" or "if not r then not s"

Theorem 3.2.1: Negation of a Universal Statement

"V- x in D, Q(x)" -> "E' x in D such that ~Q(x)" or equivalently, ~(V-x in D, Q(x)) _= E' x in D such that ~Q(x).

Definition: Universal Conditional Statement

"V- x, if P(x) then Q(x)."

Definition: Necessary and Sufficient Condition

"r if and only if s"

Order of Quantifiers

"∀ people x, ∃ people y such that x loves y." = Everyone loves everyone. "∃ people y such that ∀ people x, x loves y." = There's a person that everyone loves. Be careful about the orientation of ∀ and ∃.

Necessary and Sufficient Conditions for Universal Statements

"∀x, r(x) is a sufficient condition for s(x)" means "∀x, if r(x) then s(x)" "∀x, r(x) is a necessary condition for s(x)" means "∀x, if ~ r(x) then ~s(x)" "∀x, r(x) only if s(x)" means "∀x, if r(x) then s(x)" or "∀x, if ~ s(x) then ~ r(x)"

Examples of negating Multiple Quantify Statements

"∃ a real number u such that ∀ real numbers v, uv = v" -> S: There is a real number u that when multiplied by any real number v results in v. N: No real number has that property that its product with every real number equals that number. "∀r C- Q, ∃ integers a and b such that r = a/b" - > S: Every rational number r is the quotient of integers a and b. N: There is a rational number r that isn't the quotient of any two integers.

Cafeteria line example:

"∃ an item I such that ∀ students S, S chose I" -> There is an item that every student chose it. "∃ a student S such that ∀ items I, S chose I" -> There is a student that chose every items. "∃ a student S such that ∀ stations I, ∃ an item I in Z such that S chose I" -> There is a student that picked up at least one item at each station. "∀ students S and ∀ stations Z, ∃ an item I in Z such that S chose I" -> Each student picked an item at each station.

Examples of fallacies or errors

1. Converse Error: p -> q, q, therefore p. 2. Inverse Error: p -> q, ~p, therefore ~q.

Order of operations for logical operators

1. Evaluate everything within the parentheses first. 2. Evaluate the negations. 3. Evaluate the V /\ second. 4. Evaluate the -> and <-> third.

Rules of Inference

1. Generalization: p, therefore p V q or q, therefore p V q. 2. Specialization: p /\ q, therefore p or p/\ q, therefore q. 3. Elimination: p V q, ~q, therefore p or p V q, ~p, therefore q. 4. Transitivity: p -> q, q -> r, therefore p -> r.

Testing an argument for validity.

1. Identify the premises and conclusion of the argument form. 2. Construct a truth table showing the truth values of all the premises and conclusions. 3. A row of the truth table in which all the premises are true is called a critical row. If the conclusion within a critical row, then that means that there exists an argument of a given form that has false premises, yet a true conclusion. This means the argument is invalid.

Murder Mystery: Something wicked this way comes to Downtown Abbey. Lady Edith Crawley was found murdered in her bed Thursday morning. So far the police have discerned the following facts. O = O'brien killed Edith. T = Thomas killed Edith. H = Hughs killed Edith. F = Edith fired O'brien. SS = Edith discovered shocking truth about Hughes CC = Thomas is O'B's conspirator. CAR = Carson discovered secret.

1. O V T V H. 2. F 0+ SS. (exclusive or) 3. O -> CC _= ~CC -> ~O 4. F -> O 5. Car. 6. Car -> ~SS. Process: 6 5 Modus Ponens. (7) 2 (7) By elimination (8). 4 (8) Modus Ponens (9). 3 (9) CC. Modus Ponens (10).

Definition: Predicate

A sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of a variable.

Definition: Statement

A sentence that is true or false, but not both. Examples include: One plus one equals 2 or one plus two equals 3. False examples: His name is Jeff (Who's "His"), X+Y = 1 (What's x or y).

Definition: Argument

A sequence of statements and an argument form is a sequence of statement forms. All statements within an argument are referred to as premises, except for the final one, which is referred to as the conclusion. For an argument form to be valid, then regardless for the statements inputted, if the premises are true then the conclusion is true as well.

Definition: Existential statement.

A statement of the form "E'x C- D such that Q(x)." It is defined to be true if, and only if, Q(x) is true for at least one value of x in D. It is false if, and only if, Q(x) is false for all values of x in D.

Definition: Universal Statement

A statement of the form "V- x C- D, Q(x)." It is defined to be true if, and only if, Q(x) is true for every x in D. It is destined to be false if, and only if, Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement.

Definition: Contradiction.

A statement, in which regardless of what truth values are inputted for the statement variables, is always false. A statement whose form is a contradiction is referred to as a contradictory statement.

Definition: Tautology

A statement, in which regardless of what truth values are inputted for the statement variables, is always true. A statement whose form is a tautology is referred to as a tautological statement.

Definition: Truth set

All elements of D that make P(x) true when they are substituted for x. It is denoted as { x C- D I P(x) }.

Definition: Syllogism

An argument form that consists of two premises and a conclusion. The first and second premises are referred to as the major and minor premises respectively.

Definition: Sound and Unsound

An argument is sound if and only if it's valid as well as all it's premises are true. An argument that doesn't fit this requirement is called an unsound argument.

Definition: Fallacy

An error in reasoning that results in an invalid argument. Three common fallacies are using ambiguous premises (Treating them as factual), circular reasoning (Treating a conclusion as true without deriving from the premises), and jumping to conclusions.

Definition: Statement form or Proportional form

An expression made up of statement variables and logical connectives that becomes a statement when actual statements are substituted in. A truth table displays the values of various combinations of component statements.

Definition: Contrapositive, Converse, Inverse.

CP: if ~q then ~p. C: if q then p. I: if ~p then ~q. Conditional and Contrapositive are logically equivalent. Converse and Inverse are logically equivalent.

Definition: Variants of Universal Condition Statements

Contrapositive: ∀x, if ~Q(x) then ~P(x) Converse: ∀x, if Q(x) then P(x) Inverse: ∀x, if ~P(x) then ~Q(x) Contrapositive logically equivalent to original statement. Original statement isn't logically equivalent to converse. Original statement isn't logically equivalent to inverse.

Definition: Biconditional

Denoted as p <--> q and it is only true if p and q have the same values while it is false if p and q have opposite values.

Modus ponens

If P then Q P Therefore Q

Modus tellons

If P then Q ~Q Therefore ~P

Definition: Condition

If p and q are statement variables, the conditional of q by p is "if p then q" or "p implies q" and is denoted by p -> q. It is false if p is true and q is false. p is referred to as the hypothesis while q is referred to as the conclusion. In the event of the conditional statement being true while the hypothesis is false, it is referred to as vacuously true/true by default. An equivalent form of p -> q is ~p V q and the negation form of p -> q is p /\ ~q

Definition: Conjunction

If p and q are statement variables, the conjunction of p and q is denoted as p /\ q. It is only true if both p and q are true. Example of phrases that mean conjunction are and, but.

Definition: Disjunction

If p and q are statement variables, the disjunction of p and q is denoted as p V q. It is true if p or q is true. Examples of phrases that mean disjunction are not, neither, nor. Inequalities are also examples of disjunction (X _< 5 means X = 5 or X < 5).

Definition: Negation

If p is a statement variable, the negation of p is "Not p" or "It is not the case that p" and is denoted ~p.

Definition: Contradiction Rule

If you can show the supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true. This form of argument is symbolically represented by- ~p -> c, therefore p.

Definition: Interpreting Statements with Two Different Quantifiers

If you want to establish the truth of a statement of the form. "∀x in D, ∃y in E such that P(x,y)" Which means by any x in D, there must be a y in E that works for P(x,y). In contrast, there is "∃x in D such that ∀y in E, P(x,y)" Which means that there is an element x in D that works for every y in E for P(x,y).

Exclusive OR (XOR)

This refers to (p V q) /\ ~ (p /\ q).

Definition: Logically equivalent

Two quantified statements are logically equivalent means that the statements always have the same truth values regardless of what predicates are substituted for the predicate symbols and no matter what sets are used for the domains of the predicate variables.

Definition: Logically Equivalent

Two statements are logically equivalent if they share the exact same truth values for every possible substitution. Denoted with a _=. Checking for logical equivalency is as simple as creating a truth table for two statement variables and seeing if they are the same across each row.

Definition: Quantifiers

Words that refer to quantities such as "some" or "all" and tell how many elements a given predicate is true.

Symbols for statements:

ZZ = integers. Q = rational. RR = all reals. ∀ = For every, for each, or all. ∃ = There is a.

Absorption laws

p∨(p∧q)≡p p∧(p∨q)≡p

Method of exhaustion

showing the truth of the predicate separately for each individual element of the domain

Definition: Negations of Multiple Quantify Statements

~(∀x in D, ∃y in E such that P(x,y)) -> ∃x in D such that ∀y in E, ~ P(x,y). ~(∃x in D such that ∀y in E, P(x,y)) -> ∀x in D, ∃y in E such that ~P(x,y) ~(∀x in D, [∃y in E such that P(x,y)]) _= ∃x in D I ~ [∃y in E such that P(x,y)] -> ∃x in D I ∀y in E, ~ P(x,y)

De Morgan's laws

¬(p ∧ q) ≡ ¬p ∨ ¬q ¬(p ∨ q) ≡ ¬p ∧ ¬q

Definition: Vacuously truth of Universal Statements

∀x, if P(x) then Q(x) is called vacuously true or true by default if, and only if, P(x) is false for every x in D.

Negation of a Universal Conditional Statement

∼(∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) and ∼Q(x).