Show that the relation 'S' in set `A={x inz:0lexle12}` given by S={(a,b):a.bin z,|a-b|`is divisble by a} is an equivalence relation.

Show that each of the relation R in the set A={x in Z: 0 le xle12} ,given by 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.

Show that each of the relation R in the set A={x in Z: 0lt= xlt=12} ,given by R = {(a,b) : a=b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Show that the relation R in the set A = (1,2,3,4,5} given by R = {(a,b) : |a - b| is even }, is an equivalence relation. Show that all the elements of {1,3,5} are related to each other and all the elements of {2,4} are related to each other. But no element of {1, 3, 5} is related to any element of {2,4}.

Show that the relation R in the set is given by A={x in Z,0lexle12} R={(a,b):a-b is a multiple of 4} is an equivalence relation.

In the set of all natural numbers,let a relation defined by {(a,b):a,binN,a-b is divisible by 5} .Prove that R is an equivalence relation.

Show that the relation R on the set of natural numbers defined as R:{(x,y):y-x is a multiple of 2} is an equivalance relation

Show that the relation R in the set A of all the book in a library of a college, given by R = {(x,y): x and y have the same number of pages} is an equivalence relation.

Let f: X rarr Y , be a function. Define a relation R in X given by: R={(a, b): f(a)=f(b)} Examine if R is an equivalence relation.

Let A be the set of all students of a boys school. Show that the relation R in A given by R ={(a, b): a is sister of b} is the empty relation and R' ={(a, b): the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R on the set N xx N defined by (a, b)R(c, d) iff a d(b+c)=b c(a+d) is an equivalence relation.