Step by step
The question
Question: make the truth table of (p ∧ q) → r.
1 Rows for p, q and r
Each letter can be true or false.
Three letters give 2 × 2 × 2 = 8 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: two T, two F, and again.
r: 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 rows 1 and 2.
3 Column for (p ∧ q) → r
We read the arrow as if, then.
(p ∧ q) → r is a promise: if p ∧ q, then r.
It is broken only when p ∧ q is true and r is false.
So it is F only in row 2.
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) → r is true.
Top to bottom: T, F, T, T, T, T, T, T.
4 Check
Let's check row 2 by hand.
In row 2, p is true, q is true and r is false.
Put them in: (T ∧ T) → F.
The inside first.
It gives F, as in the table.
Right!
Let's look back.
First: rows for p, q and r.
Then: column for p ∧ q.
Then: column for (p ∧ q) → r.
Last: check.