2.2 Conditional Statements
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A conditional statement is the sentence “if p then q”, denoted symbolically p → q. 3 p is called the hypothesis. 4 q is called the conclusion.
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2022/2023
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2.2 Conditional Statements
Conditional Statements
Definition
(^1) When making a logical inference, one reasons from a hypothesis to a conclusion.
Conditional Statements
Definition
(^1) When making a logical inference, one reasons from a hypothesis to a conclusion. (^2) A conditional statement is the sentence “if p then q”, denoted symbolically p → q. (^3) p is called the hypothesis. (^4) q is called the conclusion. (^5) p → q is false when p is true and q is false, and true otherwise.
Truth Table for p → q
Fact
The truth table for p → q is
p q p → q T T
Truth Table for p → q
Fact
The truth table for p → q is
p q p → q T T T T F F F T
Truth Table for p → q
Fact
The truth table for p → q is
p q p → q T T T T F F F T T F F
Example
Example
A conditional statement that is true by virtue of the fact that its hypothesis is false is called vacuously true. For example:
If 0 = 1 then 0 = 2.
If-Then as Or
Example
(^1) Show that p → q ≡∼ p ∨ q.
If-Then as Or
Example
(^1) Show that p → q ≡∼ p ∨ q. (^2) Rewrite the following statement in an if-then form: “Either you do not get 90% or you get an A”. (^3) Show that ∼ (p → q) ≡ p∧ ∼ q.
The Contrapositive of a Conditional Statement
Definition
The contrapositive of a conditional statement of the form “If p then q” is
The Contrapositive of a Conditional Statement
Definition
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.
Conditional statement ≡ its contrapositive
Fact
A conditional statement is logically equivalent to its contrapositive.
The Converse and Inverse of a Conditional Statement
Definitions
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
The Converse and Inverse of a Conditional Statement
Definitions
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.”