Question

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d(b-c)=b c(a-d)$ Then $R$ is

• A. transitive but neither reflexive nor symmetric
• B. symmetric but neither reflexive nor transitive
• C. symmetric and transitive but not reflexive
• D. reflexive and symmetric but not transitive

Hint:

For relations, always check the properties like symmetry, reflexivity, and transitivity by testing them individually with examples.

Answer 1

The Correct Option is B

$\text{(a, b)}R(c,d)\Rightarrow ad(b-c)=bc(a-d)$
Symmetric:
$(c,d)R(a,b)\Rightarrow cb(d-a)=da(c-b)\Rightarrow$
Symmetric Reflexive:
(a, b) $R(a,b)\Rightarrow ab(b-a)\ne ba(a-b)\Rightarrow$
Not reflexive
Transitive: $(2,3)R(3,2)$ and $(3,2)R(5,30)$ but
$((2,3),(5,30))\notin R\Rightarrow \text{Not transitive}$

Answer 2

Step 1: Symmetric: \[ (c, d) \, R \, (a, b) \quad \Rightarrow \quad cb(d - a) = da(c - b) \quad \Rightarrow \quad \text{Symmetric} \] Step 2: Reflexive: \[ (a, b) \, R \, (a, b) \quad \Rightarrow \quad ab(b - a) \neq ba(a - b) \quad \text{Not reflexive} \] Step 3: Transitive: \[ (2, 3) \, R \, (3, 2) \quad \text{and} \quad (3, 2) \, R \, (5, 30) \quad \Rightarrow \quad ((2, 3), (5, 30)) \notin R \quad \text{Not transitive} \]