Exam #1

Question 1

Mathematical Statement

Answer 1

a declarative sentence that is either true or false, but not both.

Question 2

When is P→Q false?

Answer 2

The only case in which P→Q is false is if P is true, but Q is false.

Question 3

vacuously true

Answer 3

true by default

Question 4

Even number definition

Answer 4

An integer a is even provided that there exists an
integer n such that a = 2n

Question 5

Odd number definition

Answer 5

An integer a is odd provided that there exists an integer k such that
a = 2k + 1

Question 6

Negation

Answer 6

¬P

Question 7

Conjunction

Answer 7

P ∧ Q

Question 8

Disjunction

Answer 8

P ∨ Q

Question 9

Conditional

Answer 9

P → Q

Question 10

Biconditional

Answer 10

P ⇔ Q
P if and only if Q

Question 11

P ⇔ Q is equivalent to

Answer 11

(Q→P) ∧ (P→Q)

Question 12

tautology

Answer 12

A tautology is a compound statement S that is true for
all possible combinations of truth values of the component
statements that are part of S.

Question 13

Contradiction

Answer 13

A contradiction is a compound statement S that is false
for all possible combinations of truth values of the component
statements that are part of S

Question 14

Double Negation

Answer 14

¬(¬P) ≡ P

Question 15

DeMorgan’s Laws

Answer 15

¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬(P ∨ Q) ≡ ¬P ∧ ¬Q

Question 16

Negating Conditional Statements

Answer 16

¬(P → Q) ≡ P ∧ ¬Q

Question 17

Is the negation of a conditional statement also a conditional statement?

Answer 17

The negation of a conditional statement is not a
conditional statement.

Question 18

Contrapositive

Answer 18

Given a conditional statement
P→Q
the contrapositive of the statement is
¬Q → ¬P

Question 19

Contrapositive Equivalence

Answer 19

A conditional statement is logically equivalent to its
contrapositive.

Question 20

Converse

Answer 20

The converse of a conditional statement:
P→Q is Q → P

Question 21

Converse Equivalence

Answer 21

A common mistake is to assume that a statement is logically
equivalent to its converse. This is not true.

Question 22

Commutative

Answer 22

P ∧ Q ≡ Q ∧ P
P ∨ Q ≡ Q ∨ P

Question 23

Associative

Answer 23

(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)
(P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R)

Question 24

Distributive

Answer 24

P∧(Q∨R)≡(P∧Q)∨(P∧R)
P∨(Q∧R)≡(P∨R)∧(P∨Q)

Question 25

Identity

Answer 25

P∧t≡P P∨c≡P

Question 26

Negation

Answer 26

P∨¬P≡t P∧¬P≡c

Question 27

Double Negation

Answer 27

¬(¬P)≡P

Question 28

Idempotence

Answer 28

P∧P≡P P∨P≡P

Question 29

Universal bound

Answer 29

P∨t≡t P∧c≡c

Question 30

Negations of t and c

Answer 30

¬t≡c ¬c≡t

Question 31

Implication

Answer 31

P → Q ≡ ¬P ∨ Q

Question 32

Negation of a conditional

Answer 32

¬(P→ Q) ≡ P ∧ ¬Q

Question 33

Contrapositive

Answer 33

P → Q ≡ ¬Q → ¬P

Question 34

Biconditional statement

Answer 34

P ↔ Q ≡ (P → Q) ∧ (Q → P)

Question 35

Set

Answer 35

a well-defined collection of objects that can be thought of as a single entity itself

Question 36

Element

Answer 36

one of the objects in a set

Question 37

Subset

Answer 37

A is called a subset of B, written A ⊆ B, if and only if every element of A is also an element of B.

Question 38

Set Equity

Answer 38

Two sets are equal if and only if they have exactly the same elements.

Question 39

Variable

Answer 39

A variable is a symbol representing an unspecified object that can be chosen from a given set U

Question 40

Universal set for the variable

Answer 40

It is the set of specified objects from which objects may be chosen to substitute for the variable.

Question 41

Constant

Answer 41

A constant is a specific member of the universal set

Question 42

open sentence (or predicate)

Answer 42

An open sentence (or predicate) is a sentence P(x1, x2,· · · , xn) that contains a finite number of variables and becomes a true or false statement when specific values are substituted for the variables.

Question 43

Truth Sets

Answer 43

If P(x) is a predicate with one variable, the truth set of P(x) is the set of all elements in the universal set that can be substituted for x to make P(x) true

Question 44

The Universal quantifier

Answer 44

∀ (for all, for every, for each, given any, ...)

Question 45

The Existential quantifier:

Answer 45

∃ (there exists, there is a, there is at least one, for some, for at least one, ···)

Question 46

Negating Universal Statement

Answer 46

¬((∀x∈U) (P(x)))≡(∃x∈U)¬(P(x))

Question 47

The negation of (∀x ∈U)(P(x) →Q(x))

Answer 47

(∃x ∈ U) ¬(P(x) → Q(x))