What are the equivalence classes of these bit strings forthe equivalence relation in Exercise 11? a) 010b) 1011c) 11111d) 01010101

d) 01010101

[010]_{R} = the first three bits of s are 010 [1011]_{R} = the first three bits of s are 101 [11111]_{R} = s ∈ A [01010101]_{R} = the first three bits of s are 010

Explanation:

We say ∼ is an equivalence relation on a set A if it satisfies the following three properties:

a) reflexive: for all a∈A, a∼a.

b) symmetry: for all a, b∈A, if a∼b then b∼a.

c) transitivity: for all a, b, c∈A, if a∼b and b∼c then a∼c.

Given

A = Set of all bit strings that have a length of at least 3

R = { (x, y) first three bit of x and y are same

a) the equivalence classes of 010 will contain all bit strings that have as the first three digits 010

[010]_{R} = the first three bits of s are 010

b) the equivalence classes of 1011 will contain all bit strings that have as the first three digits 101

[1011]_{R} = s ∈ A

c) the equivalence classes of 11111 will contain all bit strings that have as the first three digits 111

[11111]_{R} = the first three bits of s are 111

d) the equivalence classes of 01010101 will contain all bit strings that have as the first three digits 010

[01010101]_{R} = s ∈ A