Test 1 Review

Question 1

Which Law of Logical Equivalency?
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q

Answer 1

De Morgan’s Laws

Question 2

State the Identity Laws

Answer 2

p ∧ T ≡ p
p ∨ F ≡ p

Question 3

What is the order of operations when simplifying complex logical expressions?

Operators:
¬ “not
∨ “or”
➜ “implies”
↔ “if and only if”
∧: “and”

Answer 3

1. ¬ “not
2. ∧: “and”
3. ∨ “or”
4. ➜ “implies”
5. ↔ “if and only if”

Question 4

Which Law of Logical Equivalency?
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

Answer 4

Associative Laws

Question 5

State the Logical Equivalences for Each Biconditional Statement
p ↔ q ≡
p ↔ q ≡
p ↔ q ≡
¬( p ↔ q ) ≡

Answer 5

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

Question 6

State the Idempotent Laws

Answer 6

p ∨ p ≡ p
p ∧ p ≡ p

Question 7

List 8 common ways to express conditionals in English, p ➜ q, where p is the hypothesis and q is the conclusion.

Answer 7

1. if p, then q
2. q if p
3. q follow from p
4. p implies q
5. q is necessary for p
6. p is sufficient for q
7. p only if q
8. q unless ¬p

Question 8

State the Double Negation Law

Answer 8

¬(¬p) ≡ p

Question 9

State the Converse of “If I study, then I get an A on my exam.”

Answer 9

Converse: if q, then p
q ➜ p

If I get an A on my exam, I studied.

Question 10

Which Law of Logical Equivalency?
p ∨ ¬p ≡ T
p ∧ ¬p ≡ F

Answer 10

Negation Laws

Question 11

Which Law of Logical Equivalency?
p ∧ T ≡ p
p ∨ F ≡ p

Answer 11

Identity Laws

Question 12

State the Converse, Inverse, Contrapostive, and Biconditional of if p, then q.
p ➜ q

Answer 12

Converse: if q, then p
q ➜ p

Inverse: if not p, then not q
¬p ➜ ¬q

Contrapositive: if not q, then not p
¬q ➜ ¬p

Biconditional: p if and only if q
p ↔ q

Question 13

State the Associative Laws

Answer 13

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

Question 14

State the Commutative Laws

Answer 14

p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p

Question 15

State the De Morgan’s Laws

Answer 15

¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q

Question 16

State the Absorption Laws

Answer 16

p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p

Question 17

Which Law of Logical Equivalency?
p ∨ q ≡ q ∨ p
p ∧ q ≡ q ∧ p

Answer 17

Commutative Laws

Question 18

State the Logical Equivalences for Each Conditional Statement
p ➜ q ≡
p ➜ q ≡
p ∨ q ≡
p ∧ q ≡
¬( p ➜ q ) ≡
( p ➜ q ) ∧ (p ➜ r ) ≡
( p ➜ r ) ∧ (q ➜ r ) ≡
( p ➜ q ) ∨ (p ➜ r ) ≡
( p ➜ r ) ∨ (q ➜ r ) ≡

Answer 18

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

Question 19

State the Negation Laws

Answer 19

¬p ∨ ¬p ≡ T
p ∧ ¬p ≡ F

Question 20

State the Domination Laws

Answer 20

p ∨ T ≡ T
p ∧ F ≡ F

Question 21

Which Law of Logical Equivalency?
p ∨ T ≡ T
p ∧ F ≡ F

Answer 21

Domination Laws

Question 22

Which Law of Logical Equivalency?
p ∨ p ≡ p
p ∧ p ≡ p

Answer 22

Idempotent Laws

Question 23

Which Law of Logical Equivalency?
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p

Answer 23

Absorption Laws

Question 24

Which Law of Logical Equivalency?
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Answer 24

Distributive Laws

Question 25

State the Distributive Laws

Answer 25

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

Question 26

Which Law of Logical Equivalency? ¬(¬p) ≡ p

Answer 26

Double Negation

Question 27

What is a Tautology?

Answer 27

A logical expression which is always true regardless of the truth values of the variables because it contains all possible outcomes. p ∨ ¬p

Question 28

What is a Contradiction?

Answer 28

A logical expression which is always false regardless of the truth value of the variables. It is self-contradicting. p ∧ ¬p

Question 29

What is a Contingency?

Answer 29

A logical expression which is neither a Tautology or Contradiction. p

Question 30

Modus Tollens

Answer 30

p ➜ q ¬q ---------- ∴ ¬p ( ¬q ∧ ( p ➜ q )) ➜ ¬p

Question 31

Hypothetical Syllogism

Answer 31

p ➜ q q ➜ r ---------- ∴ p ➜ r (( p ➜ q ) ∧ ( q ➜ r )) ➜ ( p ➜ r )

Question 32

Disjunctive Syllogism

Answer 32

p v q ¬p ---------- ∴ q (( p v q ) ∧ ¬p ) ➜ q

Question 33

Resolution

Answer 33

p v q ¬p v r ---------- ∴ q v r (( p v q ) ∧ ( ¬p v r )) ➜ q v r

Question 34

Addition

Answer 34

p ---------- ∴ p v q p ➜ ( p v q )

Question 35

Simplification

Answer 35

p ∧ q ---------- ∴ p ( p ∧ q ) ➜ p

Question 36

Conjunction

Answer 36

p q ---------- ∴ p ∧ q ( p ∧ q ) ➜ ( p ∧ q )

Question 37

Modus Ponens

Answer 37

p ➜ q p ---------- ∴ q ( p ∧ ( p ➜ q)) ➜ q

Question 38

When True? When False? ∀x∀y P(x, y) ∀y∀x P(x, y)

Answer 38

True: P(x, y) is true for every pair x, y. False: There is a pair x , y for which P(x,y) is false.

Question 39

When True? When False? ∀x∃y P(x, y)

Answer 39

True: For every x there is a y for which P(x,y) is true. False: There is an x such that P(x,y) is false for every y.

Question 40

When True? When False? ∃x∀y P(x, y)

Answer 40

True: There is an x for which P(x,y) is true for every y. False: For every x there is a y for which P(x,y) is false.

Question 41

When True? When False? ∃x∃y P(x, y) ∃y∃x P(x, y)

Answer 41

True: There is a pair x,y for which P(x,y) is true. False: P(x,y) is false for every pair x,y.

Question 42

Universal Specification / Instantiation (US or UI)

Answer 42

∀xP(x) ---------- ∴ P(c) ♡ for any c in the domain Explanation: The universal quantifier ∀xP(x) states that P(x) is true for all x in the domain. Therefore, you can select any c (a particular element in the domain) and conclude that P(c) is true because the statement applies universally.

Question 43

Universal Generalization (UG)

Answer 43

P(c) ---------- ∴ ∀xP(x) ♡ for an arbitrary c, not a particular one Explanation: Here, P(c) holds for a specific c, but the conclusion ∀xP(x) requires P(x) to hold for every x in the domain. This inference is only valid if c is chosen arbitrarily and represents any element in the domain (not a specific, particular one). If c were a specific element, this conclusion would be invalid.

Question 44

Existential Specification / Instantiation (ES or EI)

Answer 44

∃xP(x) ---------- ∴ P(c) ♡ for some specific c (unknown) Explanation: The existential quantifier ∃xP(x) states that there exists at least one x such that P(x) is true. Form this, you can infer that there is some specific c (though it may not be explicitly identified) such that P(c) holds true. This c satisfies P(x), but we don't know exactly which one it is.

Question 45

Existential Generalization (EG)

Answer 45

P(c) ---------- ∴ ∃xP(x) ♡ finding one c such that P(c) Explanation: If P(c) holds for a specific element c, you can conclude that ∃xP(x), since you've identified at least one instance c in the domain where P(x) is true. This c serves as a witness to the truth of the existential quantifier ∃xP(x).

Question 46

True or False? Inverse and converse are logically equivalent.

Answer 46

True

Question 47

True or False? It is possible to define disjunction using only negations and conjunction.

Answer 47

True ¬( ¬p ∧ ¬q )

Question 48

True of False? Domain-restricted universal quantification is logically equivalent to the unrestricted universal quantification of a conditional.

Answer 48

True

Question 49

True of False? Domain-restricted existential quantification is logically equivalent to the unrestricted existential quantification of a conditional.

Answer 49

False It is equal to the existential quantification of a conjunction.

Question 50

The negation of a conjunction is the disjunction of the negations.

Answer 50

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