Show that p → q ≡ q̅ → p̅
Solution
Sure, let's prove that p → q ≡ q̅ → p̅ using truth tables.
Step 1: Define the variables
Let's define p and q as two logical statements. p → q is the implication that if p is true, then q is also true. q̅ → p̅ is the implication that if not q is true, then not p is also true.
Step 2: Construct the truth table
The truth table for p, q, p → q, q̅, p̅, and q̅ → p̅ is as follows:
| p | q | p → q | q̅ | p̅ | q̅ → p̅ |
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | F | T | F | T |
| F | T | T | F | T | T |
| F | F | T | T | T | F |
Step 3: Compare the columns
We can see that the column for p → q is identical to the column for q̅ → p̅. Therefore, we can conclude that p → q ≡ q̅ → p̅.
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