Step by step
The question
Question: make the truth table of p ∧ ¬p.
1 Rows for p
Each letter can be true or false.
One letters give 2 = 2 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.
2 Column for ¬p
We read this sign as not.
¬p is the opposite of p: T becomes F, F becomes T.
So it is T only in row 2.
3 Column for p ∧ ¬p
We read this sign as and.
p ∧ ¬p is true only when p and ¬p are both true.
It is F in every row.
Careful!
And needs both.
One false makes p ∧ q false.
The last column is the answer: when p ∧ ¬p is true.
Top to bottom: F, F.
It is false in every row.
A statement that is always false is called a contradiction.
4 Check
Let's check row 2 by hand.
In row 2, p is false.
Put them in: F ∧ ¬F.
The inside first.
It gives F, as in the table.
Right!
Let's look back.
First: rows for p.
Then: column for ¬p.
Then: column for p ∧ ¬p.
Last: check.