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 this sign as or.
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 row 4.
3 Column for q ∨ p
q ∨ p 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 4.
4 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, T, T, F and T, T, T, F.
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.
5 Check
Let's check row 2 by hand.
In row 2, p is true and q is false.
The first: T ∨ F gives T.
And the second: F ∨ T gives T.
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 ∨ p.
Then: compare the columns.
Last: check.