Archangel Macsika
Write each of the following three statements in the symbolic form and determine which pairs are logically equivalent: a) If it walks like a duck and it talks like a duck, then it is a duck. b) Either it does not walk like a duck or it does not talk like a duck, or it is a duck. c) If it does not walk like a duck and it does not talk like a duck, then it is not a duck.
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Solution:
Let:
p = it walks like a duck
q = it talks like a duck
r = it is a duck
Then given statements can be represented as:
a. (p ^ q) → r
b. ~p V ~q V r
c. (~p ^ ~q) → ~r
The truth tables for each of the above statements are as below:
a. (p ^ q) → r
| p | q | r | p ^ q | (p ^ q) → r |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | T | F | F | T |
| F | F | T | F | T |
| F | F | F | F | T |
b. ~p V ~q V r
| p | q | r | ~p | ~q | ~p V ~q | ~p V ~q V r |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | T |
| T | T | F | F | F | F | F |
| T | F | T | F | T | T | T |
| T | F | F | F | T | T | T |
| F | T | T | T | F | T | T |
| F | T | F | T | F | T | T |
| F | F | T | T | T | T | T |
| F | F | F | T | T | T | T |
c. (~p ^ ~q) → ~r
| p | q | r | ~p | ~q | ~r | ~p ^ ~q | (~p ^ ~q) → ~r |
|---|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F | T |
| T | T | F | F | F | T | F | T |
| T | F | T | F | T | F | F | T |
| T | F | F | F | T | T | F | T |
| F | T | T | T | F | F | F | T |
| F | T | F | T | F | T | F | T |
| F | F | T | T | T | F | T | F |
| F | F | F | T | T | T | T | T |
Since the truth tables for statements a and b are same the two statements are equivalent.