In discrete mathematics, conditional statements (p → q) are central to logical reasoning. From a single conditional statement, we can derive three closely related forms:
Understanding these forms is essential for:
A conditional statement is written as:
p → q
“If p, then q”
p: antecedent (hypothesis)q: consequent (conclusion)The converse of a conditional statement is obtained by interchanging the antecedent and consequent.
q → p
“If q, then p”
The inverse is obtained by negating both parts of the conditional.
¬p → ¬q
“If not p, then not q”
The contrapositive is obtained by:
p and q¬q → ¬p
“If not q, then not p”
Let:
p: A number is divisible by 4q: The number is evenp → q
“If a number is divisible by 4, then it is even”
q → p
“If a number is even, then it is divisible by 4” ❌ (not always true)
¬p → ¬q
“If a number is not divisible by 4, then it is not even” ❌ (not always true)
¬q → ¬p
“If a number is not even, then it is not divisible by 4” ✔ (always true)
| p | q | p → q | q → p | ¬p → ¬q | ¬q → ¬p |
|---|---|---|---|---|---|
| True | True | True | True | True | True |
| True | False | False | True | True | False |
| False | True | True | False | False | True |
| False | False | True | True | True | True |
p → q ≡ ¬q → ¬p
✔ A conditional is logically equivalent to its contrapositive
q → p ≡ ¬p → ¬q
✔ Converse and inverse are logically equivalent to each other ❌ But NOT equivalent to the original statement
p → q and ¬q → ¬p always have the same truth valuesq → p and ¬p → ¬q always have the same truth valuesp → qq → p¬p → ¬q¬q → ¬pp → q ≡ ¬q → ¬pq → p ≡ ¬p → ¬q