Question

using truth table verify that ~(pvq)=~p^~q​

Answer 1

Answer:

p q ∼p ∼q (p∨q) ∼(p∨q) (∼p∧∼q)

T T F F T F F

T F F T T F F

F T T F T F F

F F T T F T T

∴∼(p∨q)≡∼p∧−∼q

Hence proved.

Answer 2

Answer:

It is verified that ~(p \/ q) = ~p /\ ~q​.

Explanation:

Given: ~(p \/ q) = ~p /\ ~q​

To find: Verification of the given equivalence.

Steps need to be done:

1. Write the given equivalence.
2. Find the number of rows in the truth table.
3. Find the truth table for the left-hand side.
4. Find the truth table for the right-hand side.
5. Check whether their last columns are equal.
6. Determine the equivalence.

Step 1:

Finding the number of rows in the truth table.

There are two variables given p and q.

Hence, n = 2.

Thus, the number of rows will be,

$2^{n} =2^{2}=4$

There will be four rows in the truth table.

The elements of p are T, T, F, and F.

The elements of q are T, F, T, and F.

Step 2:

Finding the truth table for ~(p \/ q).

The truth table for ~(p \/ q) is given below in the first image.

The last column comes out to be T, F, F, and F.

Step 3:

Finding the truth table for ~p /\ ~q​.

The truth table for ~p /\ ~q​ is given below in the first image.

The last column comes out to be T, F, F, and F.

Step 4:

The last columns of both tables are the same.

Thus, they are equivalent.

Final answer:

Hence, ~(p \/ q) = ~p /\ ~q is verified.