→ Implication(→):
Given two propositions p and q, the implication p→q is the proposition that is false when p is true and q is false, true otherwise.
•Here p is called hypothesis and q is called consequence.
We use different terminologies to express p→q like:
• if p , then q
• q is the consequence of p
• p only if q
Example:
if we have propositions p=Today is sunday. and q=It is hot today. then implication of p and q (p→q ): If today is sunday then it is hot day.
or Today is sunday only if it is hot today.
Truth Table
| p | q | p→q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
• “p” and “q” represent the truth values of the hypothesis and conclusion.
• “p→q” represents the truth value of the implication.
→Some of the related implications formed from p→q are:
• Inverse of implication
• Converse of implication
• Contrapositive of implication
→ Inverse of implication:
When we add “not” to the hypothesis(p) and consequence(q) of implication p→q then it becomes ⁓p→⁓q.
Example:
p=It is raining. q=The road is muddy.
implication(p→q)=If it is raining then the road is muddy.
Inverse(⁓p→⁓q)=If it is not raining then the road is not muddy.
Truth Table
| p | q | ⁓p | ⁓q | ⁓p→⁓q |
| T | T | F | F | T |
| T | F | F | T | T |
| F | T | T | F | F |
| F | F | T | T | T |
• “p” and “q” represent the truth values of the hypothesis and conclusion.
• “⁓p and ⁓q” represents the negative of p and q.
• “⁓p→⁓q” represents the truth value of the inverse of the implication.
→ Converse of implication:
When we interchange/flip the hypothesis(p) and conclusion(q) of implication p→q then the result becomes q→p which is known as converse of implication.
Example:
Implication(p→q): If it is raining then the road is muddy.
Converse(q→p): If the road is muddy then it is raining.
Truth Table
| p | q | p→q | q→p |
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
• “p” and “q” represent the truth values of the hypothesis and conclusion.
• “p → q” represents the truth value of the original implication.
• “q → p” represents the truth value of the converse of the implication.
→ Contrapositive of implication:
The contrapositive of an implication involves negating both the hypothesis (p) and the consequence (q) of the implication i.e(⁓p→⁓q) and then switching their positions to ⁓q→⁓p.
Example:
Implication(p → q): If it is raining then the road is muddy.
Contrapositive(⁓q→⁓p): If the road is not muddy then it is not raining.
Truth Table
| p | q | ⁓p | ⁓q | ⁓q→⁓p |
| T | T | F | F | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
• “p” and “q” represent the truth values of the hypothesis(p) and consequence(q).
• “p → q” represents the truth value of the original implication.
• “~q → ~p” represents the truth value of the contrapositive of the implication.
