Step by step
The question
Question: make the truth table of ¬(p ∧ q).
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.
Careful!
The ¬ is for the whole bracket, not only the first letter.
The last column is the answer: when ¬(p ∧ q) is true.
Top to bottom: F, T, T, T.
4 Check
Let's check row 1 by hand.
In row 1, p is true and q is true.
Put them in: ¬(T ∧ T).
The inside first.
It gives F, as in the table.
Right!
Let's look back.
First: rows for p and q.
Then: column for p ∧ q.
Then: column for ¬(p ∧ q).
Last: check.