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
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 5, 6, 7 and 8.
3 Column for q ∨ r
We read this sign as or.
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 4 and 8.
4 Column for ¬p ∧ (q ∨ r)
We read this sign as and.
¬p ∧ (q ∨ r) is true only when ¬p and q ∨ r are both true.
So it is T only in rows 5, 6 and 7.
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: F, F, F, F, T, T, T, F.
5 Check
Let's check row 1 by hand.
In row 1, p is true, q is true and r is true.
Put them in: ¬T ∧ (T ∨ T).
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.
Then: column for q ∨ r.
Then: column for ¬p ∧ (q ∨ r).
Last: check.