Step by step
The question
Question: are p → q and ¬p → ¬q 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 ¬p and ¬q
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 3 and 4.
¬q is the opposite of q too.
4 Column for ¬p → ¬q
¬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 3.
5 Compare the columns
Now compare the two last columns.
p → q in red, ¬p → ¬q in purple.
Row by row: the same, until row 2.
In row 2, one is F and the other is T.
One row is enough: they are not the same.
This is the inverse: not on both parts, not swapped.
It does not mean the same.
Careful!
Swapping the two parts of an if-then changes its meaning.
If it rains, the street is wet.
A wet street does not mean it rained.
So they are not equivalent.
Row 2 shows it.
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: ¬T → ¬F gives T.
Different, as the table says.
Right!
Let's look back.
First: rows for p and q.
Then: column for p → q.
Then: column for ¬p and ¬q.
Then: column for ¬p → ¬q.
Then: compare the columns.
Last: check.