# Use truth tables to verify the associative laws

To prove the associative laws, we need to use the truth table for conjunctions and disjunctions.

## Truth tables for conjunction and disjunctions

Now we can start solving the exercise.

## a) (p∨q)∨r ≡p∨(q∨r)

As in previous examples, let’s examine what will be the structure of the truth table we must create.

There are three propositional variables, so we will have 8 rows.

Also, we have four compound propositions, so we need 7 (3 variables plus 4 compound propositions) columns.

Notice that to find the truth value in each compound proposition, we need to use the truth tables for conjunction or disjunctions. For instance, in the first row pq has value T, because according the truth table for the disjunction of two propositions, TvT is T.

According to the table above, (p∨q)∨r is logically equivalent to p∨(q∨r) because they have the same truth values.

## b) (p∧q)∧r ≡p∧(q∧r)

Following a similar reasoning to the previous exercise, we can conclude that we need a table with 7 columns and 8 rows.

Notice that to find the truth value in each compound proposition, we need to use the truth tables for the conjunction of two propositions. For instance, in the third row pq has value F, because according the truth table for the disjunction of two propositions, TF is F.

In the same way, in the first row, p∧ (q∧r) has the value T because, in the first row, p is T and q∧r is also T, then TT is T according to the truth table for the conjunction of two propositions.

According to the table above, (pq)r is logically equivalent to p (qr) because they have the same truth values.

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