Question

If A and B are the set of even integers and set of odd integers, respectively, find( A union B) and (A union B) ^c

Answer 1

Answer on Question #67463 – Math – Discrete Mathematics

Question

If $A$ and $B$ are the set of even integers and set of odd integers, respectively, find $A \cup B$ and $(A \cup B)^c$.

Solution

The set of even integers formally is

$$A = \{n \mid n = 2k, \text{where } k \in \mathbb{Z}\},$$

and the set of odd integers formally is

$$B = \{n \mid n = 2k + 1, \text{where } k \in \mathbb{Z}\}.$$

Their union is

$$A \cup B = \{n \mid n \in A \text{ or } n \in B\} = \{n \mid n = 2k \text{ or } n = 2k + 1, \text{where } k \in \mathbb{Z}\}.$$

Since every integer $n$ is either even or odd, every integer $n$ belongs to the union: $n \in A \cup B$, thus the union contains the set of all integers:

$$\mathbb{Z} \subseteq A \cup B.$$

On the other hand, both $A$ and $B$ are subsets of $\mathbb{Z}$, therefore their union is a subset of $\mathbb{Z}$ as well:

$$A \cup B \subseteq \mathbb{Z}.$$

These two inclusions imply the equality:

$$A \cup B = \mathbb{Z}$$

That is, the union of $A$ and $B$ is the set of all integers.

Then the complement of the union consists of non-integer numbers:

$$(A \cup B)^c = \{n \mid n \notin A \cup B\} = \{n \mid n \notin \mathbb{Z}\}.$$

Answer:

$$A \cup B = \mathbb{Z} \text{ and } (A \cup B)^c = \{n \mid n \notin \mathbb{Z}\}.$$

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