Step by step
The question
Question: are (p ∧ q) → r and (p → r) ∧ (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 ∧ 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.
4 Column for p → r
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.
5 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.
6 Column for (p → r) ∧ (q → r)
(p → r) ∧ (q → r) is true only when p → r and q → r are both true.
So it is F only in rows 2, 4 and 6.
7 Compare the columns
Now compare the two last columns.
(p ∧ q) → r in red, (p → r) ∧ (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 gives T.
And the second: (T → F) ∧ (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 ∧ q.
Then: column for (p ∧ q) → r.
Then: column for p → r.
Then: column for q → r.
Then: column for (p → r) ∧ (q → r).
Then: compare the columns.
Last: check.