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 and ¬q
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.
¬q is the opposite of q too.
3 Column for ¬p → ¬q
We read the arrow as if, then.
¬p → ¬q is a promise: if ¬p, then ¬q.
It is broken only when ¬p is true and ¬q is false.
So it is F only in row 3.
Careful!
If the if-part is false, the promise is not broken.
So F → F is T, and so is F → T.
The last column is the answer: when ¬p → ¬q is true.
Top to bottom: T, T, F, T.
4 Check
Let's check row 3 by hand.
In row 3, p is false and q is true.
Put them in: ¬F → ¬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 and ¬q.
Then: column for ¬p → ¬q.
Last: check.