Implication transformations
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Implication transformations
Implication can be expressed using other logical operators. These transformations allow p → q to be rewritten in equivalent forms.
Statements
- p → q ≡ ¬p ∨ q
- ¬(p → q) ≡ p ∧ ¬q
- p ↔ q ≡ (p → q) ∧ (q → p)
Explanation
- The implication p → q is equivalent to "not p or q".
- The negation of p → q is equivalent to "p and not q".
- Equivalence p ↔ q can be defined using two implications.
Examples
- "If it rains then the ground is wet"
≡ "Either it does not rain, or the ground is wet".
- "It is not the case that (if I study then I pass)"
≡ "I study and I do not pass".
Truth Table (p → q ≡ ¬p ∨ q)
| p | q | p → q | ¬p ∨ q |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |