JEE Main Relations Question | Symmetric or Transitive? PYQ

 

❓ Question

Let

$$A=\{0,1,2,3,4,5\}$$

and let $R$ be a relation on $A$ defined by

$$(x,y)\in R \iff \max(x,y)\in\{3,4\}.$$

Consider the statements:

S1: Number of elements in $R$is 18.

S2: Relation $R$ is symmetric but neither reflexive nor transitive.

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✍️ Solution

S1: Number of elements in $R$

Case 1: $\max(x,y)=3$

Then,

• $x\le3,\; y\le3$
  
• At least one of them is 3.

Possible ordered pairs:

$$(3,0),(3,1),(3,2),(3,3), (0,3),(1,3),(2,3)$$

Total pairs:

$$7$$

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Case 2: $\max(x,y)=4$

Then,

• $x\le4,\; y\le4$
  
• At least one of them is 4.

Possible ordered pairs:

$$(4,0),(4,1),(4,2),(4,3),(4,4), (0,4),(1,4),(2,4),(3,4)$$

Total pairs:

$$9$$

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Therefore,

$$|R|=7+9=16$$

Hence,

$$\boxed{\text{S1 is False}}$$

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S2: Nature of the relation

(i) Reflexive

A relation is reflexive if

$$(x,x)\in R\quad\forall x\in A.$$

But,

$$(0,0):\quad \max(0,0)=0\notin\{3,4\}$$

Hence,

$$\boxed{R\text{ is not reflexive}.}$$

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(ii) Symmetric

Suppose

$$(x,y)\in R.$$

Then,

$$\max(x,y)\in\{3,4\}.$$

Since

$$\max(x,y)=\max(y,x),$$

we also have

$$(y,x)\in R.$$

Therefore,

$$\boxed{R\text{ is symmetric}.}$$

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(iii) Transitive

Take

$$(1,3)\in R,\qquad (3,0)\in R$$

because

$$\max(1,3)=3,\qquad \max(3,0)=3.$$

But,

$$(1,0)\notin R$$

since

$$\max(1,0)=1\notin\{3,4\}.$$

Thus,

$$\boxed{R\text{ is not transitive}.}$$

Hence,

$$\boxed{\text{S2 is True}.}$$

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✅ Final Answer

• S1: ❌ False
• S2: ✅ True