1. Logic ChAPTER 3

2. Variations of the Conditional and Implications 3.4

3. Variations of the conditional
Variations of p → q
Converse:
Inverse:
Contrapositive:
p → q is logically equivalent to ~ q → ~ p
q → p is logically equivalent to ~ p → ~ q

4. Examples of variations
p: n is not an even number.
q: n is not divisible by 2.
Conditional: p → q
If n is not an even number, then n is not divisible by 2.
Converse: q → p
If n is not divisible by 2, then n is not an even number.
Inverse: ~ p → ~ q
If n is an even number, then n is divisible by 2.
Contrapositive: ~q → ~p
If n is divisible by 2, then n is an even number.

5. p q T F p → q T F q → p T F ~p → ~ q T F ~q → ~p T F Equivalent
Conditional
Converse
Inverse
Contrapositive p q T F
p → q T F
q → p T F
~p → ~ q T F
~q → ~p T F

6. Conditional Equivalents
Statement
Equivalent forms
if p, then q p → q
p is sufficient for q
q is necessary for p
p only if q
q if p

7. Biconditional Equivalents
Statement
Equivalent forms
p if and only if q
p ↔ q
p is necessary and sufficient for q
q is necessary and sufficient for p
q if and only if p

8. Examples of variations
Given: h: honk
u: you love Ultimate
Write the following in symbolic form.
Honk if you love Ultimate.
If you love Ultimate, honk.
Honk only if you love Ultimate.
A necessary condition for loving Ultimate is to honk.
A sufficient condition for loving Ultimate is to honk
To love Ultimate, it is sufficient and necessary that you honk. or

9. Tautologies and Contradictions
A statement that is always true is called a tautology. A statement that is always false is called a contradiction.

10. Example
Show by means of a truth table that the statement p ↔ ~ p is a contradiction. p ~p
p ↔ ~ p T F

11. Implications
The statement p is said to imply the statement q, p  q, if and only if the conditional p → q is a tautology.

12. Example
Show that p q
p → q
(p → q) Λ p
[(p → q) Λ p] → q T F
END