First slide

Relations XII

Question

Let X = {1,2,3, 4, 5, 6} and Y = {1,3,5,7,9}. Which of the following is/are relations from X to Y?

Easy

A

$R_{1} = \{(x, y) : y = 2 + x, x \in X, y \in Y\}$
B

$R_{2} = \{(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)\}$
C

$R_{3} = \{(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)\}$
D

$R_{4} = {(1, 3), (2, 5), (2, 4), (7, 9)$

Solution

X = {1,2,3, 4,5}
Y = {1,3, 5, 7, 9}
(a). $R_{1} = {(x, y) : y = 2 + x, x \in X, y \in Y}$
$\begin{array}{ll} x = 1 & y = 3 \\ x = 2 & y = 4 \\ x = 3 & y = 5 \\ x = 4 & y = 6 \\ x = 5 & y = 7 \end{array}$
So, R is not a relation from X to Y.
(b).
$\begin{aligned} R_{2} = {(1, 1), (2, 1), (3, 3) (4, 3), (5, 5)} \\ R_{2} \subseteq X \times Y \end{aligned}$
(c).
$\begin{aligned} R_{3} = {(1, 1), (1, 3), (3, 5) (3, 7) (5, 7)} \\ R_{3} \subseteq X \times Y \end{aligned}$
(d). $R_{4} ⊈ X \times Y$

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