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Definition of Relations

Let A= 1,2,...

Question

Let $A = {1, 2, 3, 4, 5}, B = {1, 3, 5, 7, 9, 10, 17, 26, 30}$ Which of the following sets of ordered pairs are (i) relations (ii) functions (iii) neither,from A to B
(a) $R_{1} = {(1, 3), (3, 3), (5, 17), (1, 2)}$
(b) $R_{2} = {(3, 7), (4, 5), (5, 30)}$
(c) $R_{3} = {(x, y) : x ϵ A, y ϵ B, x + y = 8}$
(d) $R_{4} = {(x, y) : x ϵ A, y ϵ B, y = x^{2} + 1}$ (e) $R_{5} = {(x, y) : x ϵ A, y ϵ B, y = 2 x - 1}$

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Solution

$R_{1} \neq R$ as $(1, 2) ϵ A \times B,$
$R_{1} \neq f$ as $2$ or $4 ϵ A$ has no image in B,
$R_{2} = R$ but $R_{2} \neq f$ as in $R_{1}$
$R_{3} = {(1, 7), (3, 5), (5, 3)}$
$x + y = 8, x ϵ A, y ϵ B$
$∴ R_{3} = R$ being a subset of $A \times B$ but $R_{3} \neq f$ as in $R_{1}$
$R_{4} = {(2, 5), (3, 10) (4, 17), (5, 26)}$
$∵ y = x^{2} + 1, x ϵ A, y ϵ B$
$R_{4} = R$ but $R_{4} \neq f$ as in $R_{1}$
$R_{5} = {(1, 1), (2, 3), (3, 5), (4, 7), (5, 9)}$
$y = 2 x - 1$
$R_{5} = R$ and $R_{5} = f$ as each element $ϵ A$ has a unique image in B.

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Similar questions

A = 1, 2, 3, 4, 5

B = 1, 3, 5, 7, 9, 10, 17, 26, 30 .

R_{1}

R_{2}

R_{3} = (x, y) : x \in A, y \in B, x + y = 8

R_{4} = (x, y) : x \in A, y \in B, y = x^{2} + 1

R_{5} = (x, y) : x \in A, y \in B, y = 2 x - 1

X = {1, 2, 3, 4, 5}

Y = {1, 3, 5, 7, 9}

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

R_{2}

= {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}

R_{3}

= {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}

R_{4}

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

(x, y) : x ϵ A, y ϵ B a n d x \geq y

X = {1, 2, 3, 4, 5}

Y = {1, 3, 5, 7, 9}

X \to Y

= [- 1, 1]

= [0, 1]

= [- 1, 0]

S_{1} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ A}

S_{2} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ B}

S_{3} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ C}

S_{4} = {(x, y) | x^{2} + y^{2} = 1, x ϵ B, y ϵ C}

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