Mathematical Reasoning
Implications
1. The meaning of implications "If","Only if","if and only if". A sentence with if p, then q can be written in the following ways.
(i) p implies q (denoted by p ⇒ q)
(ii) p is a sufficient condition for q
(iii) q is a necessary condition for p
(iv) ∼q implies ∼p
2. The contrapositive of a statement p ⇒ q is a statement ∼q ⇒ ∼p. The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q.
Implications: Part 1
Implications: Part 2
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1. Truth table
| p | q | p ⇒ q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
2.
| p | q | p ⇔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
- Law of converse : p ⇒ q converse q ⇒ p
- Law of Inverse : p ⇒ q Inverse ∼p ⇒ ∼q
- Law of Contrapositive : p ⇒ q Contrapositive ∼q ⇒ ∼p