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Relations And Functions

Question
CBSEENMA12032259
Wired Faculty App

If R is the relation in N x N defined by (a, b) R (c, d) if and only if a + d = b + c, show that R is equivalence relation. 

Solution

Here (a, b) R (c, d) ⇔ a + d = b + c.
(i) Now (a, b) R (a, b) if a + b = b + a, which is true.
∴ relation R is reflexive.
(ii) Now (a, b) R (c, d)
⇒ a + d = b + c ⇒ d + a = c + b
⇒ c + b = d + a ⇒ (c, d) R (a, b)
∴ relation R is symmetric.
(iii) Now (a, b) R (c, d) and (c, d) R (e,f)
⇒ a + d = b + c and c + f = d + e
⇒ (a + d) + (c + f) = (b + c) + (d + e) ⇒ a + f = b + e
⇒ (a , b) R (e, f)
∴ relation R is transitive.
Now R is reflexive, symmetric and transitive
∴ relation R is an equivalence relation.

Some More Questions From Relations and Functions Chapter

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’. 

If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

Let A = {1. 2. 3}. Then show that the number of relations containing (1,2) and (2. 3) which are reflexive and transitive but not symmetric is four.

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