Skip to main content

Beginning Activity Beginning Activity 2: Congruence Modulo 3
A4US

An important equivalence relation that we have studied is congruence modulo n on the integers. We can also define subsets of the integers based on congruence modulo n. We will illustrate this with congruence modulo 3. For example, we can define C[0] to be the set of all integers a that are congruent to 0 modulo 3. That is,

C[0]={a∈Z∣a≡0(mod3)}.

Since an integer a is congruent to 0 modulo 3 if and only if 3 divides a, we can use the roster method to specify this set as follows:

C[0]={…,−9,−6,−3,0,3,6,9,…}.

1.

Use the roster method to specify each of the following sets:

(a)

The set C[1] of all integers a that are congruent to 1 modulo 3. That is, C[1]={a∈Z∣a≡1(mod3)}.

(b)

The set C[2] of all integers a that are congruent to 2 modulo 3. That is, C[2]={a∈Z∣a≡2(mod3)}.

(c)

The set C[3] of all integers a that are congruent to 3 modulo 3. That is, C[3]={a∈Z∣a≡3(mod3)}.

2.

Now consider the three sets, C[0], C[1], and C[2].

(a)

Determine the intersection of any two of these sets. That is, determine C[0]∩C[1], C[0]∩C[2], and C[1]∩C[2].

(b)

Let n=734. What is the remainder when n is divided by 3? Which of the three sets, if any, contains n=734?

(c)

Repeat Task 2.b for n=79 and for n=−79.

(d)

Do you think that C[0]∪C[1]∪C[2]=Z? Explain.

(e)

Is the set C[3] equal to one of the sets C[0],C[1], or C[2]?

(f)

We can also define C[4]={a∈Z∣a≡4(mod3)}. Is this set equal to any of the previous sets we have studied in this part? Explain.