Q.Let $A = \{x \in Z : 0 \le x \le 12\}$. Show that $R = \{(a,b) : a, b \in A,\ |a - b|\ \text{is divisible by 4}\}$ is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class $\{2\}$. OR Show that function $f : R \to R$ defined by $f(x) = \dfrac{x}{x^2 + 1},\ \forall\, x \in R$ is neither one-one nor onto. Also, if $g : R \to R$ is defined as $g(x) = 2x - 1$, find $fog\,(x)$.

Step 1 — The key principle. Every equivalence relation on a set is reflexive, symmetric and transitive, and there is a one-to-one correspondence between equivalence relations on $A$ and partitions of $A$ (the blocks are the equivalence classes). So instead of listing relations, count partitions.

Step 2 — Impose the condition. The relation must contain $(b, c)$. Hence $b \sim c$, so $b$ and $c$ must lie in the same equivalence class.

Step 3 — Enumerate the partitions of $\{a,b,c\}$ in which $b, c$ share a block.

1. $\{\{a\},\ \{b, c\}\}$ — gives the relation $R_1 = \{(a,a),(b,b),(c,c),(b,c),(c,b)\}.$ …

Concept & Intuition

An equivalence relation is a way of saying two things are "the same" in some abstract sense. To be a proper notion of sameness, it must obey three rules:

• Reflexive: Everything is the same as itself.
• Symmetric: If A is the same as B, then B is the same as A.
• Transitive: If A is the same as B and B is the same as C, then A is the same as C.

We test each given relation against these three properties. If any one fails, it is not an equivalence relation.

Step-by-step analysis

1. Option (A): $a R_1 b \iff |a| = |b|$

   • Reflexive: For any real $a$, $|a| = |a|$ always holds. ✓
   • Symmetric: If $|a| = |b|$, then clearly $|b| = |a|$. ✓
   • Transitive: If $|a| = |b|$ and $|b| = |c|$, then $|a| = |c|$. ✓ All three properties hold, so this is an equivalence relation. The equivalence classes are sets of numbers with the same absolute value (e.g., $\{2, -2\}$).

2. Option (B): $a R_3 b \iff a \text{ divides } b$ (on real numbers)

   • Reflexive: Does $a$ divide $a$? For real numbers, "divides" usually means $b/a$ is an integer. For $a=0$, division is undefined; for $a \neq 0$, $a/a = 1$ is an integer, so it holds for nonzero $a$. But reflexivity must hold for every real number. Since $0$ does not divide itself (division by zero is undefined), reflexivity fails. ✗ (Even if we restrict to nonzero reals, symmetry fails: $2$ divides $4$ but $4$ does not divide $2$.)

3. Option (C): $a R_2 b \iff a \geq b$

   • Reflexive: $a \geq a$ is true for all $a$. ✓ …

Concept: equivalence classes of the relation 'congruent modulo 5' on the whole numbers.

aRb iff a and b leave the same remainder on division by 5, i.e. a = b (mod 5).

The equivalence class of 1 is

[1] = { w in W : w = 1 (mod 5) } = { 1, 6, 11, 16, 21, ... }. …