Animated Solution for Mathematics - Sets and Relations: If the Boolean expression $(p\oplus q)\wedge (\sim p\odot q)$ is equivalent to $p\wedge q$, where $\oplus ,\odot \in \{\wedge ,\vee \}$, then the ordered pair $(\oplus ,\odot )$ is:

We are tasked with finding the Boolean operators $\oplus$ and $\odot$ such that the expression $(p\oplus q)\wedge (\sim p\odot q)$ is logically equivalent to the intersection $p\wedge q$.

The target expression $p\wedge q$ represents the region where both $p$ and $q$ are true. Our objective is to identify the specific operators that reduce the given compound statement to this intersection.

Let us test the candidate operators: $\oplus =\wedge$ and $\odot =\vee$. Substituting these into the original expression, we obtain:

To simplify this, we apply the Distributive Law. We distribute the term $(p\wedge q)$ over the disjunction $(\sim p\vee q)$ as follows:

We now evaluate the two resulting terms independently to determine their truth values.

The First Term: $((p\wedge q)\wedge \sim p)$. By applying the Associative Law, we rearrange this to $(p\wedge \sim p)\wedge q$. Since $p\wedge \sim p$ is a contradiction (always False), the entire term simplifies to $F\wedge q$, which is $F$.

The Second Term: $(p\wedge q)\wedge q$. Using the Associative Law, we rewrite this as $p\wedge (q\wedge q)$. By the Idempotent Law, $q\wedge q$ simplifies to $q$, leaving us with $p\wedge q$.

Combining these results, the expression becomes $F\vee (p\wedge q)$.

By the Identity Law, the disjunction of False with any expression is the expression itself. Thus, the entire statement simplifies to:

This matches our target perfectly. We have confirmed that the ordered pair $(\wedge ,\vee )$ is the correct solution.