Solution to Write each of the following three statements in the symbolic form and determine which pairs … - Sikademy

Oct. 28, 2020

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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.

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

pqrp ^ q(p ^ q) → r
TTTTT
TTFTF
TFTFT
TFFFT
FTTFT
FTFFT
FFTFT
FFFFT

b. ~p V ~q V r

pqr~p~q~p V ~q~p V ~q V r
TTTFFFT
TTFFFFF
TFTFTTT
TFFFTTT
FTTTFTT
FTFTFTT
FFTTTTT
FFFTTTT

c. (~p ^ ~q) → ~r

pqr~p~q~r~p ^ ~q(~p ^ ~q) → ~r
TTTFFFFT
TTFFFTFT
TFTFTFFT
TFFFTTFT
FTTTFFFT
FTFTFTFT
FFTTTFTF
FFFTTTTT

Since the truth tables for statements a and b are same the two statements are equivalent.

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