Step by step
The question
Question: are (p → r) ∨ (q → r) and (p ∨ q) → r equivalent?
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 → r
We read the arrow as if, then.
p → r is a promise: if p, then r.
It is broken only when p is true and r is false.
So it is F only in rows 2 and 4.
3 Column for q → r
q → r is a promise: if q, then r.
It is broken only when q is true and r is false.
So it is F only in rows 2 and 6.
4 Column for (p → r) ∨ (q → r)
We read this sign as or.
(p → r) ∨ (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 2.
5 Column for p ∨ q
p ∨ q 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 7 and 8.
6 Column for (p ∨ q) → r
(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 rows 2, 4 and 6.
7 Compare the columns
Now compare the two last columns.
(p → r) ∨ (q → r) in red, (p ∨ q) → r in purple.
Row by row: the same, until row 4.
In row 4, one is T and the other is F.
One row is enough: they are not the same.
Careful!
To be equivalent they must match in every row.
One row that differs is enough to say no.
So they are not equivalent.
Row 4 shows it.
8 Check
Let's check row 4 by hand.
In row 4, p is true, q is false and r is false.
The first: (T → F) ∨ (F → F) gives T.
And the second: (T ∨ F) → F gives F.
Different, as the table says.
Right!
Let's look back.
First: rows for p, q and r.
Then: column for p → r.
Then: column for q → r.
Then: column for (p → r) ∨ (q → r).
Then: column for p ∨ q.
Then: column for (p ∨ q) → r.
Then: compare the columns.
Last: check.