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

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.