Let a set A = A1 [] A2 [] .....

Question

Let a set A = A1 $$\cup$$ A2 $$\cup$$ ..... $$\cup$$ Ak, where Ai $$\cap$$ Aj = $$\phi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relation R from A to A by R = {(x, y) : y $$\in$$ Ai if and only if x $$\in$$ Ai, 1 $$\le$$ i $$\le$$ k}. Then, R is :

Accepted Answer

$$R = \{ (x,y):y \in {A_i},\,iff\,x \in {A_i}\,1 \le i \ge k\} $$

(1) Reflexive

(a, a) $$\Rightarrow$$ $$a \in {A_i}$$ iff $$a \in {A_i}$$

(2) Symmetric

(a, b) $$\Rightarrow$$ $$a \in {A_i}$$ iff $$b \in {A_i}$$

(b, a) $$\in$$R as $$b \in {A_i}$$ iff $$a \in {A_i}$$

(3) Transitive

(a, b) $$\in$$R \& (b, c) $$\in$$R.

$$\Rightarrow$$ $$a \in {A_i}$$ iff $$b \in {A_i}$$ \& $$b \in {A_i}$$ iff $$c \in {A_i}$$

$$\Rightarrow$$ $$a \in {A_i}$$ iff $$c \in {A_i}$$

$$\Rightarrow$$ (a, c) $$\in$$ R.

$$\Rightarrow$$ RElation is equivalnece.

Solutions · 1