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
We read this sign as not.
¬p is the opposite of p: T becomes F, F becomes T.
So it is T only in rows 3 and 4.
3 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 3.
Careful!
And needs both.
One false makes p ∧ q false.
The last column is the answer: when ¬p ∧ q is true.
Top to bottom: F, F, T, F.
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.
Then: column for ¬p ∧ q.
Last: check.