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 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 rows 3 and 4.
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 row 4.
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, T, T, F, T, T, T, T.
4 Check
Let's check row 4 by hand.
In row 4, p is true, q is false and r is false.
Put them in: (T → F) ∨ 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.