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i) Which type of relation is shown in below expression R1 = { (a,b) | a = b }

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Solution:

R_1 = \{ (a,b) | a = b \}

So, \{(a,a)\}\in R_1 , thus it is reflexive relation.

Next,

(a,b)\in R_1 \Rightarrow (b,a)\in R_1 \\ \because a=b \\\Rightarrow (a,a)\in R_1 \Rightarrow (a,a)\in R_1

thus it is symmetric relation.

Further,

(a,b)\in R_1, (b,c)\in R_1 \Rightarrow (a,c)\in R_1 \\ \because a=b \\\Rightarrow b=c, a=c \\\Rightarrow (a,a)\in R_1, (a,a)\in R_1 \Rightarrow (a,a)\in R_1

thus it is transitive relation.

Hence, it is an equivalence relation.


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