Congruence modulo 3 relation partitions the set Z into how many equivalence classes ?

Prove that the relation 'Congruence modulo, m' on the set Z of all integers is an equivalence relation.

Suppose a box contains a set of n balls (n gt 4) (denoted by B )of four different colours (many have different sizes), viz,red, blue, green and yellow. Show that a relation R defined on B as R={(b_1,b_2) : "balls" b_1 "and" b_2 have the same colour} is an equivalence relation on B. How many equivalence classes can you find with respect ot R ?

If R and S are two equivalence relation on the set then prove that RcapS is also an equivlaence relation on the set.

Prove that a-=b mod 3 is an equivalence relation on the set of integers Z. Find its congruence classes.

Show that AxxA is an equivalence relation on a non-empty set A. What is the corresponding equivalence class.

Write the equivalence class [3]_7 as a set.

List the members of the equivalence relation defined by {{1,2,3,4}} partitins on X={1,2,3,4}.Also find the equivalence classes of 1,2,3 and 4.

Let R be a relation defined on the set of natural numbers N as follows: R = {(x,y): x in N,y in N and 2x + y = 24). Then, find the domain and range of the relation R. Also, find whether R is an equivalence relation or not.

Show that the relation R is in the set A={1,2,3,4,5} given by R={(a, b): |a-b| is divisible by 2}, is an equivalence relation. Write all the equivalence classes of R.