Animated Solution for Mathematics - Sets and Relations: Let $Z$ be the set of integers. If $A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

Welcome, my dear students! Today, we are embarking on a mathematical journey that is as elegant as it is fundamental. We are going to dissect a problem that bridges the gap between algebraic equations and the structural beauty of set theory.

Our mission is to determine the number of subsets of the Cartesian product of two sets, $A$ and $B$. By the end of this, you will see the logic flowing clearly through each step.

Let us first turn our attention to Set $A$, defined by the equation:

We know that any non-zero number raised to the power of zero is exactly $1$. Therefore, for this equation to hold true, the exponent must be zero:

Setting the linear factor $(x+2)$ to zero gives us $x=-2$. For the quadratic factor, we solve $x^2-5x+6=0$, which factors into $(x-2)(x-3)=0$.

This yields two additional roots: $x=2$ and $x=3$. Since all these values are integers, our set is defined as $A=\{-2,2,3\}$.

The cardinality, or the number of elements in $A$, is $n(A)=3$. We have successfully conquered the first peak!

Now, let us look at Set $B$, defined by the compound inequality $-3<2x-1<9$. Our goal is to isolate $x$ in the center.

We start by adding $1$ to all three parts of the inequality:

Next, we divide the entire inequality by $2$. Since $2$ is a positive number, the inequality signs remain unchanged:

This tells us that $x$ must be an integer strictly between $-1$ and $5$. Listing these out, we find $x\in \{0,1,2,3,4\}$.

Counting these values, we see that the cardinality of Set $B$ is $n(B)=5$. We are making incredible progress!

We have reached the final stage of our journey. We need to find the number of subsets of the Cartesian product $A\times B$.

First, we calculate the number of elements in the Cartesian product using the formula:

Substituting our values, we get:

Now, we apply the fundamental theorem of set theory: if a set has $n$ elements, the total number of subsets is $2^n$. With $n=15$, the total number of subsets is $2^{15}$.

Look at that! We started with a complex exponential equation and a compound inequality, and through logical, step-by-step reduction, we arrived at a clean, elegant result. This is the essence of JEE Advanced mathematics.