Animated Solution for Mathematics - Sets and Relations: Which one of the following Boolean expressions is a tautology ?

In Boolean logic, a tautology is a statement that remains true regardless of the truth values of its constituent variables. While truth tables are a reliable method, they are often tedious. Instead, we utilize algebraic laws such as the Distributive Law, the Associative Law, and the Law of Excluded Middle ($p\vee \sim p=T$) to simplify expressions.

We begin with the expression:

To test for a tautology, we check the case where $p=T$ and $q=T$: 1. The first bracket becomes $(T\vee T)=T$. 2. The second bracket becomes $(\sim T\vee \sim T)=(F\vee F)=F$. 3. The total expression results in $(T\wedge F)=F$.

Since the expression evaluates to False in this instance, Option 1 is not a tautology.

Next, we examine the expression:

Using the Distributive Law, we factor out the common term $p\wedge$:

Since $(q\vee \sim q)=T$, the expression simplifies to:

Because the result depends on the value of $p$, Option 2 is not a tautology.

We now look at the expression:

Applying the Distributive Law by factoring out $p\vee$, we obtain:

Since $(q\wedge \sim q)=F$ (a contradiction), the expression simplifies to:

Again, the result is tethered to the value of $p$, so Option 3 is not a tautology.

Option 4

Finally, we evaluate the expression:

Since all operators are 'OR' ($\vee$), we apply the Associative Law to remove the brackets:

Using the Commutative Law to rearrange the terms, we group the variables with their negations:

Applying the Law of Excluded Middle, we find:

Because the expression evaluates to True for all possible values of $p$ and $q$, Option 4 is the tautology.