Chapter 2 LOGIC

Question 1

Statement

Answer 1

Statement- A statement (or proposition) is an assertion that is true or false, but not both. We often use P, Q, and R to denote statements, or perhaps P1, P2, . . ., Pn if several statements are involved.

Each statement has a truth value,

true (denoted by T) or false (denoted by F)

Question 2

Open sentence

Answer 2

Open sentence- An open sentence is an assertion that contains one or more variables and which becomes a statement when specific values are substituted for these variables. (2) An open sentence that contains a variable x is typically represented by P(x), or Q(x).

Question 3

Domain

Answer 3

Domain - The domain of an open sentence is the collection from which the value of the variable is chosen

Question 4

Truth TablE

Answer 4

Truth Table-Often we use a table to list all possible truth values of a statement. This table is called a truth table.

Question 5

Negation

Answer 5

Negation- Let P be a statement. The negation of P is the statement not P, and is denoted by ∼ P

Question 6

Disjunction

Answer 6

Disjunction- Let P and Q be statements. The disjunction of P and Q is the statement

P or Q,

and is denoted by P ∨ Q. In this case, “or” is inclusive or, that is P ∨ Q is true if at least one of P and Q is true.

Question 7

Conjunction

Answer 7

Conjunction- Let P and Q be statements. The conjunction of P and Q is the statement

P and Q,

and is denoted by P ∧ Q

Question 8

Implication

Answer 8

Implication- Let P and Q be statements. The implication (or conditional) is the statement

If P, then Q,

and is denoted by P ⇒ Q. In this case, P is called the hypothesis and Q is called the conclusion. We also express P ⇒ Q in words as P implies Q.

Let P and Q be statements. Given the implication P ⇒ Q, there are 4 possibilities:

Question 9

Converse

Answer 9

Converse- Let P and Q be statements. The implication Q ⇒ P is called the converse of P ⇒ Q.

Question 10

Biconditional

Answer 10

Biconditional- Let P and Q be statements. The biconditional of P and Q is the conjunction

(P ⇒ Q) ∧ (Q ⇒ P),

is denoted by P ⇔ Q. In this case we say

P is equivalent to Q, or P if and only if Q,

Let P and Q be statements. The truth table for implication P ⇔ Q is

Question 11

logical Connectives

Answer 11

logical Connectives- The symbols ∼, ∨, ∧, ⇒ and ⇔ are called logical connectives

Question 12

Compound Statement

Answer 12

Compound Statement- A compound statement is a statement composed of one or more given statements and at least one logical connective.

Question 13

Tautology

Answer 13

Tautology- A compound statement S is called a tautology if it is true for all possible combinations of truth values of the component statements that comprise S.

Question 14

Contradiction

Answer 14

Contradiction- A compound statement S is called a contradiction if it is false for all possible combinations of truth values of the component statements that comprise S.

Question 15

logically equivalent-

Answer 15

logically equivalent- Let R and S be two (compound) statements. We say that R and S are logically equivalent if they have the same truth table. In this case, we write R ≡ S.

Question 16

Commutative Laws

Answer 16

Question 17

Associative Laws

Answer 17

Question 18

Distributive Laws

Answer 18

Question 19

De Morgan’s Laws

Answer 19

Question 20

Universal quantifier

Answer 20

Question 21

Existential quantifier

Answer 21

Question 22

Unique existential quantifier

Answer 22

Question 23

Logical Laws

Answer 23

Question 24

Negation of Quantified Statements

Answer 24