Mathematics

Mathematics

Problem

1. Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A={1,2,3,…,13,14}\mathrm{A}=\{1,2,3, \ldots, 13,14\} defined as R={(x,y):3x−y=0}\mathrm{R}=\{(x, y): 3 x-y=0\} (ii) Relation R in the set N\mathbf{N} of natural numbers defined as R={(x,y):y=x+5 and x<4}\mathrm{R}=\{(x, y): y=x+5 \text { and } x<4\} (iii) Relation R in the set A={1,2,3,4,5,6}\mathrm{A}=\{1,2,3,4,5,6\} as R={(x,y):y is divisible by x}\mathrm{R}=\{(x, y): y \text { is divisible by } x\} (iv) Relation R in the set Z\mathbf{Z} of all integers defined as R={(x,y):x−y is an integer }\mathrm{R}=\{(x, y): x-y \text { 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\mathrm{R}=\{(x, y): x and yy work at the same place }\} (b) R={(x,y):x\mathrm{R}=\{(x, y): x and yy live in the same locality }\} (c) R={(x,y):x\mathrm{R}=\{(x, y): x is exactly 7 cm taller than y}y\} (d) R={(x,y):x\mathrm{R}=\{(x, y): x is wife of y}y\} (e) R={(x,y):x\mathrm{R}=\{(x, y): x is father of y}y\} (i) Relation R in the set A={1,2,3,…,13,14}\mathrm{A}=\{1,2,3, \ldots, 13,14\} defined as R={(x,y):3x−y=0}\mathrm{R}=\{(x, y): 3 x-y=0\} (ii) Relation R in the set N\mathbf{N} of natural numbers defined as R={(x,y):y=x+5 and x<4}\mathrm{R}=\{(x, y): y=x+5 \text { and } x<4\} (iii) Relation R in the set A={1,2,3,4,5,6}\mathrm{A}=\{1,2,3,4,5,6\} as R={(x,y):y is divisible by x}\mathrm{R}=\{(x, y): y \text { is divisible by } x\} (iv) Relation R in the set Z\mathbf{Z} of all integers defined as R={(x,y):x−y is an integer }\mathrm{R}=\{(x, y): x-y \text { is an integer }\} e. (v)(a) Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x\mathrm{R}=\{(x, y): x and yy work at the same place \} f. (v)(b) Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x\mathrm{R}=\{(x, y): x and yy live in the same locality \} g. (v)(c) Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x\mathrm{R}=\{(x, y): x is exactly 7 cm taller than y}y\} h. (v)(d) Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x\mathrm{R}=\{(x, y): x is wife of y}y\} i. (v)(e) Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x\mathrm{R}=\{(x, y): x is father of y}y\}

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