Animated Solution for Mathematics - Sets and Relations: Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by $a{R}_{1}b⟺ab\ge 0$ and $a{R}_{2}b⟺a\ge b$, then

A Journey into Equivalence

Welcome, future engineers! Today, we are going to peel back the layers of a fundamental concept in set theory: the equivalence relation.

It is essentially a way of grouping elements that "behave" the same way. To be an equivalence relation, a relation $R$ must pass three rigorous tests: Reflexivity, Symmetry, and Transitivity. If it fails even one, it is not an equivalence relation.

Imagine the Cartesian plane where the horizontal axis is $a$ and the vertical axis is $b$. The condition $ab\ge 0$ implies that $a$ and $b$ must share the same sign, or at least one of them must be zero. Geometrically, this covers the 1st quadrant (both positive) and the 3rd quadrant (both negative).

Reflexivity: Does $a{R}_{1}a$ hold? We check: Since the square of any real number is non-negative, $R_1$ is reflexive.

Symmetry: If $a{R}_{1}b$, does $b{R}_{1}a$ follow? We know $ab\ge 0$. Because multiplication is commutative, $ba=ab\ge 0$. Thus, $R_1$ is symmetric.

Transitivity: This is the trap. If $a{R}_{1}b$ and $b{R}_{1}c$, must $a{R}_{1}c$?

Let us use a "bridge" with zero. Let $a=2$, $b=0$, and $c=-2$. We see that $2R_{1}0$ is true ($2\cdot 0=0\ge 0$) and $0R_1-2$ is true ($0\cdot -2=0\ge 0$).

However, $2R_1-2$ is false because: The bridge fails! Thus, $R_1$ is not an equivalence relation.

Now, consider $R_2$, where $a\ge b$. This is a simple inequality.

Reflexivity: Is $a\ge a$? Yes, every number is equal to itself. So, $R_2$ is reflexive.

Symmetry: If $a\ge b$, does $b\ge a$? Let us test $a=2$ and $b=1$. We have $2\ge 1$, which is true. But if we swap them, we get $1\ge 2$, which is clearly false.

The relation is directional, not symmetric. Because it fails symmetry, $R_2$ cannot be an equivalence relation.

We have put both relations through the fire. $R_1$ failed transitivity, and $R_2$ failed symmetry.

Therefore, neither is an equivalence relation. This problem teaches us that we cannot rely on intuition alone; we must rigorously test every property. Keep visualizing, keep testing, and you will master these concepts in no time!