Are ~(p and q) and ~p and ~q equivalent

Step by step

The question

Question: are ¬(p ∧ q) and ¬p ∧ ¬q equivalent?

1 Rows for p and q

Each letter can be true or false.

Two letters give 2 × 2 = 4 rows: every way they can go.

We make a table: a column for each letter, then a column for each part, from the inside out.

We write T for true and F for false.

p: the first half T, the second half F.

This way no row is missed.

q: T, F, T, F, down the column.

2 Column for p ∧ q

We read this sign as and.

p ∧ q is true only when p and q are both true.

So it is T only in row 1.

3 Column for ¬(p ∧ q)

We read this sign as not.

¬(p ∧ q) is the opposite of p ∧ q: T becomes F, F becomes T.

So it is F only in row 1.

4 Column for ¬p and ¬q

¬p is the opposite of p: T becomes F, F becomes T.

So it is T only in rows 3 and 4.

¬q is the opposite of q too.

5 Column for ¬p ∧ ¬q

¬p ∧ ¬q is true only when ¬p and ¬q are both true.

So it is T only in row 4.

6 Compare the columns

Now compare the two last columns.

¬(p ∧ q) in red, ¬p ∧ ¬q in purple.

Row by row: the same, until row 2.

In row 2, one is T and the other is F.

One row is enough: they are not the same.

Careful!

When the not goes onto each part, the sign in the middle must swap.

That is De Morgan's law.

So they are not equivalent.

Row 2 shows it.

7 Check

Let's check row 2 by hand.

In row 2, p is true and q is false.

The first: ¬(T ∧ F) gives T.

And the second: ¬T ∧ ¬F gives F.

Different, as the table says.

Right!

Let's look back.

First: rows for p and q.

Then: column for p ∧ q.

Then: column for ¬(p ∧ q).

Then: column for ¬p and ¬q.

Then: column for ¬p ∧ ¬q.

Then: compare the columns.

Last: check.