Relation

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #11

A relation `R = {(x, y) : x+2y=19, x,y \in A}` defined on the set `A = {1, 2, 3, ...., 10}`. Then find `R^{-1}`.
Ans: {(17, 1), (15, 2), (13, 3), (11, 4), (9, 5), (7, 6), (5, 7), (3, 8), (1, 9)}
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Question No: #12

A relation `\rho` defined on a set `A = {1, 2, 3}` by `\rho = {(1,1), (2,3), (3,3)}`. Add minimum members to `\rho` so that `\rho` becomes an equivalence relation.

Question No: #13

Let `\mathbb{Z}` be the set of all integers. A relation `\rho` over set `\mathbb{Z}` defined by `\rho = {(x,y) : x+y \text{ is an even number, where } x,y \in \mathbb{Z}}`. Show that `\rho` is an equivalence relation.

Question No: #14

Show that the relation `\rho = {(x,y) : x\ley}` defined over the set of all real numbers `\mathbb{R}` is reflexive and transitive, but not symmetric.

Question No: #15

Show that the relation `\rho = {(x,y) \in \mathbb{Z} \times \mathbb{Z} : 3x+4y \text{ is divisible by } 7}` defined over the set of all integers `\mathbb{Z}`, is an equivalence relation.

Question No: #16

Examine that the relation `\rho = {(x,y) \in \mathbb{N} \times \mathbb{N} : x-y+\sqrt{3} \text{ is an irrational number}}` defined over the set of all natural numbers `\mathbb{N}`, is an equivalence relation or not.

Question No: #17

A relation `\rho` defined on plane `\mathbb{R}^2` by : `\rho = [ {(a,b),(c,d)} \in \mathbb{R}^2 \times \mathbb{R}^2 | a^2+b^2 = c^2+d^2 ]`. Show that the relation `\rho` is an equivalence relation on `\mathbb{R}^2`.

Question No: #18

A relation `\rho` is defined on the set of all natural numbers `\mathbb{N}` by : `(x,y) \in \rho => (x-y)` is divisible by 5 `AA x,y \in \mathbb{N}`. Prove that `\rho` is an equivalence relation on `\mathbb{N}`

Question No: #19

A relation `R` defined by : `R ={(2x-5, x-3) : x \lt 12, x` even and `x \in \mathbb{N}}`, find the domain of definition and range of `R`.
Ans: {-1, 3, 7, 11, 15}, {-1, 1, 3, 5, 7}
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