+91-8076753736[email protected]
logo
logo
-->

Relations And Functions

Question
CBSEENMA12032466
Wired Faculty App

Show that the relation R defined in the set A of all polygons as R = {(P1, P2) P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle T with sides 3, 4 and 5 ?

Solution

A is the set of all polygons
R = {(P1, P2) : P1 and P2 have same number of sides }
Since P and P have the same number of sides
∴ (P.P) ∈ R ∀ P ∈ A.
∴ R is reflexive.
Let (P1, P2) ∴ R
⇒ P1 and P2 have the same number of sides ⇒ P2 and P1 have the same number of sides ⇒ (P2, P1) ∈ R
∴ (P1, P2) ∈ R ⇒ (P2, P1) ∈ R ∴ R is symmetric.
Let (P1, P2) ∈ R and (P2, P3) ∈ R.
⇒ P1 and P2 have the same number of sides and P2 and P3 have same number of sides
⇒ P1 and P3 have the same number of sides
⇒ (P1, P3) ∈ R
∴ (P1, P2), (P2, P3) ∈ R ∈ (P1, P3) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.
Now T is a triangle.
Let P be any element of A.
Now P ∈ A is related to T iff P and T have the same number of sides P is a triangle required set is the set of all triangles in A.

Some More Questions From Relations and Functions Chapter

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

 Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’. 

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation