Name: Commutative & Associative Property of Binary Operation
Uploaded: 2019-07-19
Duration: 5 min 47 s
Description: To know more about the properties of Binary operations, watch the video and explore the different problems presented. Commit the concept to your memory with...

Question

Let $R$ be the relation defined on $Q_{0}$ (set of non-zero rational numbers) defined by $R = \{(a, b) : a, b \in Q_{0}, b = \frac{1}{a}\}$ . Check the relation $R$ for reflexivity, symmetricity and transitivity.

Accepted Answer

Given that, $R$ be the relation defined on $Q_{0}$ (set of non-zero rational numbers) defined by $R = \{(a, b) : a, b \in Q_{0}, b = \frac{1}{a}\}$ .

Clearly $(1, 1) \in R$ as $1 = \frac{1}{1}$

But $(2, 2) \in R as 2 \neq \frac{1}{2}$

$∴$ $R$ is not reflexive on $Q_{0}$ $\forall a \in Q_{0}$

Now, $(a, a) \in R \Rightarrow a = \frac{1}{a}$

$\Rightarrow a^{2} = 1 \Rightarrow a - \pm 1$

For any other ordered pair reflexivity does not hold.

Symmetry: $\forall a, b \in Q_{0}$ , $(a, b) \in R$

$\Rightarrow b = \frac{1}{a} \Rightarrow a = \frac{1}{b} \Rightarrow (b, a) \in R$

$∴$ $R$ is symmetric on $Q_{0}$ .

Transitivity : $\forall a, b, c \in Q_{0}$ , $(a, b) \in R$ and $(b, c) \in R$

Since $(a, b) \in R \Rightarrow b = \frac{1}{a}$

And $(b, c) \in R \Rightarrow c = \frac{1}{b}$

$∴ c = \frac{1}{1 / a} = a$

$c \neq \frac{1}{a} \Rightarrow (a, c) \in R$

$R$ is not transitive on $Q_{0}$ .

Clearly $(2, \frac{1}{2}) \cdot (\frac{1}{2}, 2) \in R$ but $(2, 2) \notin R$

Thus $R$ is neither reflexive nor transitive but symmetric.

Let $R$ be the relation defined on $Q_{0}$ (set of non-zero rational numbers) defined by $R = \{(a, b) : a, b \in Q_{0}, b = \frac{1}{a}\}$ . Check the relation $R$ for reflexivity, symmetricity and transitivity.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Let R be the relation defined on Q0 (set of non-zero rational numbers) defined by R=(a,b):a,b∈Q0,b=1a. Check the relation R for reflexivity, symmetricity and transitivity.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Let $R$ be the relation defined on $Q_{0}$ (set of non-zero rational numbers) defined by $R = \{(a, b) : a, b \in Q_{0}, b = \frac{1}{a}\}$ . Check the relation $R$ for reflexivity, symmetricity and transitivity.