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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 \(n\text{.}\) 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,

\begin{equation*} C[ 0 ] = \left\{ { {a \in \mathbb{Z} } \mid a \equiv 0 \pmod 3} \right\}\!\text{.} \end{equation*}

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

\begin{equation*} C[0] = \left\{ \ldots, -9, -6, -3, 0, 3, 6, 9, \ldots \right\}\text{.} \end{equation*}

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, \(C[ 1 ] = \left\{ { {a \in \mathbb{Z} } \mid a \equiv 1 \pmod 3} \right\}\!\text{.}\)

(b)

The set \(C[ 2 ]\) of all integers \(a\) that are congruent to 2 modulo 3. That is, \(C[ 2 ] = \left\{ { {a \in \mathbb{Z} } \mid a \equiv 2 \pmod 3} \right\}\!\text{.}\)

(c)

The set \(C[ 3 ]\) of all integers \(a\) that are congruent to 3 modulo 3. That is, \(C[ 3 ] = \left\{ { {a \in \mathbb{Z} } \mid a \equiv 3 \pmod 3} \right\}\!\text{.}\)

2.

Now consider the three sets, \(C[ 0 ]\text{,}\) \(C[ 1 ]\text{,}\) and \(C[ 2 ]\text{.}\)

(a)

Determine the intersection of any two of these sets. That is, determine \(C[ 0 ] \cap C[ 1 ]\text{,}\) \(C[ 0 ] \cap C[ 2 ]\text{,}\) and \(C[ 1 ] \cap C[ 2 ]\text{.}\)

(b)

Let \(n = 734\text{.}\) What is the remainder when \(n\) is divided by 3? Which of the three sets, if any, contains \(n = 734\text{?}\)

(c)

Repeat TaskĀ 2.b for \(n = 79\) and for \(n=-79\text{.}\)

(d)

Do you think that \(C[ 0 ] \cup C[ 1 ] \cup C[ 2 ] = \mathbb{Z}\text{?}\) Explain.

(e)

Is the set \(C[3]\) equal to one of the sets \(C[0], C[1]\text{,}\) or \(C[2]\text{?}\)

(f)

We can also define \(C[4] = \left\{ { {a \in \mathbb{Z} } \mid a \equiv 4 \pmod 3} \right\}\!\text{.}\) Is this set equal to any of the previous sets we have studied in this part? Explain.