Step by step
The question
Question: are p → q and ¬q → ¬p equivalent?
1 Rows for p and q
Each letter can be true or false.
Two letters give 2 × 2 = 4 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: 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 row 2.
3 Column for ¬q and ¬p
We read this sign as not.
¬q is the opposite of q: T becomes F, F becomes T.
So it is T only in rows 2 and 4.
¬p is the opposite of p too.
4 Column for ¬q → ¬p
¬q → ¬p is a promise: if ¬q, then ¬p.
It is broken only when ¬q is true and ¬p is false.
So it is F only in row 2.
5 Compare the columns
Now compare the two last columns.
p → q in red, ¬q → ¬p in purple.
Row by row they are the same: T, F, T, T and T, F, T, T.
This is the contrapositive: swap the two parts and put not on both.
It always means the same.
Careful!
To be equivalent they must match in every row.
One row that differs is enough to say no.
So they are equivalent.
We write p → q ≡ ¬q → ¬p.
6 Check
Let's check row 2 by hand.
In row 2, p is true and q is false.
The first: T → F gives F.
And the second: ¬F → ¬T gives F.
The same, as the table says.
Right!
Let's look back.
First: rows for p and q.
Then: column for p → q.
Then: column for ¬q and ¬p.
Then: column for ¬q → ¬p.
Then: compare the columns.
Last: check.