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 q ∧ r
We read this sign as and.
q ∧ r is true only when q and r are both true.
So it is T only in rows 1 and 5.
3 Column for p ∨ (q ∧ r)
We read this sign as or.
p ∨ (q ∧ r) is true when at least one of them is true.
It is false only when both are false.
So it is F only in rows 6, 7 and 8.
Careful!
Or includes both.
When both are true, T ∨ T is T.
The last column is the answer: when p ∨ (q ∧ r) is true.
Top to bottom: T, T, T, T, T, F, F, F.
4 Check
Let's check row 6 by hand.
In row 6, p is false, q is true and r is false.
Put them in: F ∨ (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 q ∧ r.
Then: column for p ∨ (q ∧ r).
Last: check.