Q.9 Show that each of the relation R in the set A = \{x \in Z : 0 \leq x \leq 12\}, given by

(i) R = \{(a, b) : |a - b|\; is\;a\;multiple \;of\; 4\} is an equivalence relation. Find the set of all elements related to 1 in each case.

Answers (1)

A = \{x \in Z : 0 \leq x \leq 12\}

A=\left \{ 0,1,2,3,4,5,6,7,8,9,10,11,12 \right \}

R = \{(a, b) : |a - b|\; is\;a\;multiple \;of\; 4\}

For a\in A , (a,a)\in R  as \left | a-a \right |=0 which is multiple of 4.

Henec, it is reflexive.

Let, (a,b)\in R i.e. \left | a-b \right | is multiple of 4.

then  \left | b-a \right | is also multiple of 4 because \left | a-b \right |  =  \left | b-a \right |  i.e.(b,a)\in R

Hence, it is symmetric.

Let, (a,b)\in R i.e. \left | a-b \right | is multiple of 4   and    (b,c)\in R  i.e. \left | b-c \right | is multiple of 4 .

 ( a-b ) is multiple of 4  and   (b-c) is multiple of 4

(a-c)=(a-b)+(b-c)  is multiple of 4

\left | a-c \right | is multiple of 4 i.e. (a,c)\in R

Hence, it is transitive.

Thus, it is reflexive,symmetric and transitive i.e. it is an equivalence relation.

The set of all elements related to 1 is  \left \{1,5,9 \right \}

\left | 1-1 \right |=0 is multiple of 4.

\left | 5-1 \right |=4 is multiple of 4.

\left | 9-1 \right |=8 is multiple of 4.

 

 

 

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