Question

If $p$: it is raining, $q$ : weather is humid, which of the following statements are logically equivalent? Justify!

• A. If it is not raining then the weather is not humid.
• B. It is raining if and only if the weather is humid.
• C. It is not true that it is not raining or the weather is humid.
• D. It is raining but the weather is not humid.

Answer 1

Answer: (C), (D)

It is raining but the weather is not humid.
It is not true that it is not raining or the weather is humid.

The symbolic forms of the statements are:

Let $P:$It is raining.

$q:$Weather is humid

Option$\left(a\right):$If it is raining then the weather is not humid.$=\sim p\to \sim q$

Option$\left(b\right):$It is raining if and only if the weather is humid.$=p\leftrightarrow q$

Option$\left(c\right):$It is not true that it is not raining or the weather is humid.$=\sim (p\vee q)$

Option$\left(d\right):$It is raining but the weather is not humid.$=p\wedge \sim q$

Truth table for all the above statements are shown below:

(a)$pq\sim p\sim q\sim p\vee \sim q\sim p\to \sim q$

$TTFFTT$

$TFFTFT$

$FTTFTF$

$FFTTTT$

(b)$pq\sim p\sim q\sim p\vee \sim qp\leftrightarrow q$

$TTFFTT$

$TFFTFF$

$FTTFTF$

$FFTTTT$

(c)$pq\sim p\sim q\sim p\vee \sim q\sim (p\vee q)$

$TTFFTF$

$TFFTFT$

$FTTFTF$

$FFTTTF$

(d)$pq\sim p\sim q\sim p\vee \sim qp\wedge \sim q$

$TTFFTF$

$TFFTFT$

$FTTFTF$

$FFTTTF$

Option $\left(c\right)$ and Option $\left(d\right)$ are same from the above table.

Hence Option$\left(c\right)$ and Option $\left(d\right)$ are logically equivalent.