MTH104 Assignment no. 1 Spring 2022 Due Date 21 - 6

ASSIGNMENT NO. 1

MTH104 SPRING 2022

DUE DATE: 21 – 6 – 2022

QUESTION NO. 1:-

Let S = {1, 2, 3, …., 19, 20}. Let $\equiv$

be the equivalence relation on S defined by the congruence modulo 7.

(a) Find the quotient set S/≡.

(b) Find a system of equivalence class representatives consisting of even integers.

SOLUTION:-

(a) Find the quotient set S/≡.

Suppose that R is an equivalence relation on a set S. For each a in S, let [a] denote the set of elements of S to which a is related under R; that is,

$\left[ a \right] = \left\{ {x:\left( {a,x} \right) \in R} \right\}$

We call [a] the equivalence class of an in S under R. the collection of all such equivalence classes is denoted by S/R, that is.

S/R $= \left\{ {\left[ a \right]:a \in S} \right\}$

It is called the quotient set of S by R.

a = b (module m)

m/a-b or m/b-a

a = b (module 7)

[ $\left( {1,1} \right),\left( {1,8} \right),\left( {1,15} \right),$ $\left( {2,2} \right),\left( {2,9} \right),\left( {2,16} \right),\left( {3,3} \right),\left( {3,10} \right),\left( {3,17} \right),\left( {4,4} \right),$ $\left( {4,11} \right),\left( {4,18} \right),\left( {5,5} \right),\left( {5,12} \right),\left( {5,19} \right),$ $\left( {6,6} \right),\left( {6,13} \right),\left( {6,20} \right),\left( {7,7} \right),\left( {7,14} \right),$ $\left( {8,1} \right),\left( {8,8} \right),\left( {8,15} \right),\left( {9,2} \right),\left( {9,9} \right),\left( {9,16} \right),$ $\left( {10,3} \right),\left( {10,10} \right),\left( {10,17} \right),\left( {11,4} \right),$ $\left( {11,11} \right),\left( {11,18} \right),\left( {12,5} \right),\left( {12,12} \right),\left( {12,19} \right),$ $\left( {13,6} \right),\left( {13,13} \right),\left( {13,19} \right),\left( {14,7} \right),$ $\left( {14,14} \right),\left( {15,1} \right),\left( {15,8} \right),\left( {15,15} \right),\left( {16,2} \right),$ $\left( {14,14} \right),\left( {15,1} \right),\left( {15,8} \right),\left( {15,15} \right),\left( {16,2} \right),$ $\left( {16,9} \right),\left( {17,3} \right),\left( {17,10} \right),\left( {17,17} \right),\left( {18,4} \right),$ $\left( {18,11} \right),\left( {18,18} \right),\left( {19,5} \right),\left( {19,12} \right),$ $\left( {19,19} \right),\left( {20,6} \right),\left( {20,13} \right)\,\,and\,\left( {20,20} \right)$ ]