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Relations and Functions
Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this
(i) On Z+, define * by a * b = a – b (ii) On Z+, define *by a * b = a b
(iii) On R , define * by a * b = a b2 (iv) On Z+, define * by a * b = | a – b |
(v) On Z+ , define * by a * b = a
(i) If a, b ∈ Z+, then a – b may or may not belong to H ; for example 3–5 = –2 ∉ Z. is not a binary operation on Z+.
(ii) If a, b ∈ Z+, then a b also belong to Z+ . is a binary operation on Z+.
(iii) If a, b ∈ R, then a b2 ∈ R.
(iv) Since | a – b | ≥ 0 , therefore, for all a, b ∈ Z+, a * b ∈ Z+. is a binary operation on Z+.
(v) For all a, b ∈ Z+ a * b = a ∈ Z+, is a binary operation on Z+.
Sponsor Area
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.
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}
Sponsor Area
Sponsor Area