MATH-2305 Ch 2 & 3

2.1 - What is a statement?

A statement (or proposition) is a sentence that is true or false but not both.

2.1 - What is a statement form?

A statement form (or propositional form) is an expression made up of statement variables (such as p,q, and r) and logical connectives (such as ∼,∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables.

2.1 - What is a tautology?

A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a tautology is a tautological statement.

2.1 - What is the negation law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Negation laws: p∨∼p≡t p∧∼p≡c

2.1 - What are the negations of t and c as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Negations of t and c: ∼t≡c ∼c≡t

2.1 - What is the universal bound law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Universal bound laws: p∨t≡t p∧c≡c

2.2 - What is a biconditional statement?

Given statement variables p and q, the biconditional of p and q is "p if, and only if, q" and is denoted p ↔ q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values. The words if and only if are sometimes abbreviated iff.

3.1 - What is the notation for the universal conditional?

Let P(x) and Q(x) be predicates and suppose the common domain of x is D. • The notation P(x) ⇒ Q (x) means that every element in the truth set of P(x) is in the truth set of Q(x), or, equivalently, ∀x, P(x) → Q(x).

2.1 - What is a contradiction?

A contradication is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a contradication is a contradictory statement.

3.1 - What is a predicate?

A predicate is 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 the variable.

3.1 - What is the universal conditional statement?

A reasonable argument can be made that the most important form of statement in mathematics is the universal conditional statement: ∀x, if P(x) then Q(x)

2.3 - What is an argument?

An argument is a collected series of statements (premises) to establish a definite proposition (conclusion). An argument is valid provided that in every circumstance in which all of the premises are true, the conclusion is also true. An argument is sound provided it is valid and all of the premises are true.

2.3 - What makes an argument sound/unsound?

An argument is called sound if, and only if, it is valid and all its premises are true. An argument that is not sound is called unsound.

3.2 - What is the converse of a universal conditional statement?

Consider a statement of the form: ∀x ∈ D, if P(x) then Q(x). Its converse is the statement: ∀x ∈ D, if Q(x) then P(x). A universal conditional statement is NOT logically equivalent to its converse.

3.2 - What is the contrapositive of a universal conditional statement?

Consider a statement of the form: ∀x ∈ D, if P(x) then Q(x). Its contrapositive is the statement: ∀x ∈ D, if ∼Q(x) then ∼P(x). A universal conditional statement is logically equivalent to its contrapositive.

3.2 - What is the inverse of a universal conditional statement?

Consider a statement of the form: ∀x ∈ D, if P(x) then Q(x). Its inverse is the statement: ∀x ∈ D, if ∼P(x) then ∼Q(x). A universal conditional statement is NOT logically equivalent to its inverse.

3.4 - What is the quantified form of the converse error?

Converse Error (Quantified Form) -Formal Version ∀x, if P(x) then Q(x). Q(a) for a particular a. ∴ P(a). ← invalid conclusion -Informal Version If x makes P(x) true, then x makes Q(x) true. a makes Q(x) true. ∴ a makes P(x) true. ← invalid conclusion

2.1 - What is the absorption law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Absorption laws: p∨(p∧q)≡p p∧(p∨q)≡p

2.1 - What is the associative law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Associative laws: (p∧q)∧r ≡ p∧(q∧r) (p∨q)∨r ≡ p∨(q∨r)

2.1 - What is the commutative law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Commutative laws: p∧q≡q∧p p∨q≡q∨p

2.1 - What is the De Morgan's laws as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. De Morgan's laws: ∼(p∧q)≡∼p∨∼q ∼(p∨q)≡∼p∧∼q

2.1 - What is the distributive law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Distributive laws: p∧(q∨r)≡(p∧q)∨(p∧r) p∨(q∧r)≡(p∨q)∧(p∨r)

2.1 - What is the double negative law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Double negative law: ∼(∼p)≡p

2.1 - What is the indempotent law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Idempotent laws: p∧p≡p p∨p≡p

2.1 - What is the identity law as applied to logical equivalences?

Given any statement variables p,q, and r, a tautology t and a contradiction c, the following logical equivalence holds. Identity laws: p∧t≡ p p∨c≡ p

3.1 - What is the truth set of a predicate?

If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set of P(x) is denoted {x ∈ D | P(x)}

2.2 - What is a conditional statement?

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 p → q. It is false when p is true and q is false; otherwise it is true. We call p the hypothesis (or antecedent) of the conditional and q the conclusion (or consequent).

2.2 - What are the necessary and sufficient conditions?

If r and s are statements: -r is a sufficient condition for s means "if r then s." -r is a necessary condition for s means "if not r then not s." -r is a necessary condition for s also means "if s then r." -Consequently, r is a necessary and sufficient condition for s means "r if, and only if, s."

3.3 - How do you interpret a statement of the form ∀x in D,∃y in E such that P(x, y)?

If you want to establish the truth of a statement of the form ∀x in D,∃y in E such that P(x, y) your challenge is to allow someone else to pick whatever element x in D they wish and then you must find an element y in E that "works" for that particular x.

3.3 - How do you interpret a statement of the form ∃x in D such that∀y in E, P(x, y)?

If you want to establish the truth of a statement of the form ∃x in D such that∀y in E, P(x, y) your job is to find one particular x in D that will "work" no matter what y in E anyone might choose to challenge you with.

3.2 - What is vacuous truth?

In general, a statement of the form ∀x in D, 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.

3.4 - What is the quantified form of the inverse error?

Inverse Error (Quantified Form) -Formal Version ∀x, if P(x) then Q(x). ∼P(a), for a particular a. ∴ ∼Q(a). ← invalid conclusion -Informal Version If x makes P(x) true, then x makes Q(x) true. a does not make P(x) true. ∴ a does not make Q(x) true. ← invalid conclusion

3.1 - What is the notation for the universal biconditional?

Let P(x) and Q(x) be predicates and suppose the common domain of x is D. • The notation P(x) ⇔ Q (x) means that P(x) and Q(x) have identical truth sets, or, equivalently, ∀x, P(x) ↔ Q(x).

3.1 - What is a universal statement?

Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form "∀x ∈ D, Q(x)." It is defined to be true if, and only if, Q(x) is true for every x in D. It is defined 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.

3.1 - What is an existential statement?

Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form "∃x ∈ D such that Q(x)." It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.

2.3 - What is modus tollens?

Modus tollens is Latin meaning "method of denying" (the conclusion is a denial). If p then q. ∼q ∴ ∼p

3.2 - How do you negate a universal conditional statement?

Negation of a Universal Conditional Statement ∼(∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) and ∼Q(x).

3.3 - How do you negate multiply-quantified statements?

Negations of Multiply-Quantified 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).

2.2 - What is the order of operations for logical operators?

Order of Operations for Logical Operators 1. ∼ - Evaluate negations first. 2. ∧,∨ - Evaluate ∧ and ∨ second. When both are present, parentheses may be needed. 3. →,↔ - Evaluate → and ↔ third. When both are present, parentheses may be needed.

2.2 - What is the converse and inverse of a conditional statement?

Suppose a conditional statement of the form "If p then q" is given. 1. The converse is "If q then p." 2. The inverse is "If ∼p then ∼q." Symbolically, The converse of p → q is q → p, The inverse of p → q is ∼p → ∼q. Note: 1. A conditional statement and its converse are not logically equivalent. 2. A conditional statement and its inverse are not logically equivalent. 3. The converse and the inverse of a conditional statement are logically equivalent to each other.

2.3 - How do you test an argument form for validity?

Testing an Argument Form 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 the conclusion. 3. A row of the truth table in which all the premises are true is called a critical row. If there is a critical row in which the conclusion is false, then it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid. If the conclusion in every critical row is true, then the argument form is valid.

2.4 - What is the Peirce Arrow?

The Peirce arrow "↓" represents the NOR gate P↓Q≡∼(P∨Q) The truth tabvle for a NOR is: P Q R=P|Q 1 1 1 1 0 0 0 1 0 0 0 0

2.4 - What is a Sheffer stroke?

The Sheffer stroke "|" represents the NAND gate. P|Q≡∼(P∧Q) The truth table for a NAND is: P Q R=P|Q 1 1 0 1 0 1 0 1 1 0 0 1

2.2 - What is the contrapositive of a conditional statement?

The contrapositive of a conditional statement of the form "If p then q" is If ∼q then ∼p. Symbolically, The contrapositive of p → q is ∼q → ∼p. The fact is that A conditional statement is logically equivalent to its contrapositive

3.4 - How do you prove validity of arguments with quantified statements?

The intuitive definition of validity for arguments with quantified statements is the same as for arguments with compound statements. An argument is valid if, and only if, the truth of its conclusion follows necessarily from the truth of its premises. The formal definition is as follows: • Definition To say that an argument form is valid means the following: No matter what particular predicates are substituted for the predicate symbols in its premises, if the resulting premise statements are all true, then the conclusion is also true. An argument is called valid if, and only if, its form is valid.

3.2 - How do you negate a universal statement?

The negation of a statement of the form ∀x in D, Q(x) is logically equivalent to a statement of the form ∃x in D such that ∼Q(x). Symbolically, ∼(∀x ∈ D, Q(x)) ≡ ∃x ∈ D such that ∼Q(x)

2.1 - What are De Morgan's Laws of logic?

The negation of an and statement is logically equivalent to the or statement in which each component is negated. ∼(p∧q)≡∼p∨∼q and The negation of an or statement is logically equivalent to the and statement in which each component is negated. ∼(p∨q)≡∼p∧∼q

3.4 - What is the rule of universal instantiation?

The rule of universal instantiation says the following: If some property is true of everything in a set, then it is true of any particular thing in the set.

3.1 - What is the universal quantifier?

The symbol ∀ denotes "for all" and is called the universal quantifier.

3.1 - What is the existential quantifier?

The symbol ∃ denotes "there exists" and is called the existential quantifier.

2.3 - What is modus ponens?

The term modus ponens is Latin meaning "method of affirming" (the conclusion is an affirmation). If p then q. p ∴q

2.1 = What is logical equivalence?

Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q. Two statements are called logically equivalent if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements.

3.4 - What is universal modus ponens?

Universal Modus Ponens -Formal Version ∀x, if P(x) then Q(x). P(a) for a particular a. ∴ Q(a). -Informal Version If x makes P(x) true, then x makes Q(x) true. a makes P(x) true. ∴ a makes Q(x) true.

3.4 - What is universal modus tollens?

Universal Modus Tollens -Formal Version ∀x, if P(x) then Q(x). ∼Q(a), for a particular a. ∴ ∼P(a). -Informal Version If x makes P(x) true, then x makes Q(x) true. a does not make Q(x) true. ∴ a does not make P(x) true.

2.2 - What is the logical equivalence of p→q?

p → q ≡ ∼p ∨ q The logical equivalence of "if p then q" and "not p or q" is occasionally used in everyday speech.

2.1 - What is the conjunction of p and q?

conjunction is represented by the symbol ∧ which denotes "and." Therefore, the conjunction of p and q is written as "p∧q" and is read "p and q." Additionally, "p but q" means "p∧q" and "neither p nor q" means "∼p∧∼q"

2.1 - What is the disjunction of p and q?

disjunction is represented by the symbol ∨ which denotes "or." Therefore, the disjunction of p and q is written as "p∨q" and is read "p or q."

2.1 - What is the negation of p?

negation is represented by the symbol ∼ which denotes "not." Therefore, the negation of p is written as "∼p" and is read "not p."

3.2 - How do the definitions of necessary, sufficient, and only if extend to universal conditional 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)" or, equivalently, "∀x, if s(x) then r(x)." • "∀x,r(x) only if s(x)" means "∀x, if ∼s(x) then ∼r(x)" or, equivalently, "∀x, if r(x) then s(x)."

2.2 - What is the negation of a conditional statement?

∼(p → q) ≡ p ∧ ∼q By definition, p → q is false if, and only if, its hypothesis, p, is true and its conclusion, q, is false. It follows that The negation of "if p then q" is logically equivalent to "p and not q."