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Chapter 1: Problem 3

Let \(B\) be any set Show that \((A \cap B) \cup C=A \cap(B \cup C)\), f and only if \(C \subseteq A\)

Short Answer

Expert verified
The given statement \((A \cap B) \cup C = A \cap (B \cup C)\) is true if and only if \(C \subseteq A\). This conclusion is reached using principles of set theory, specifically the concepts of intersection, union, and subset.

Step by step solution

01

Assume \(C \subseteq A\)

First we will assume that \(C\) is a subset of \(A\). That is, each element of \(C\) is also in \(A\).
02

Show \((A \cap B) \cup C=A \cap(B \cup C)\)

Given that \(C \subseteq A\), this means that \(A \cap B \cup C=A \cap (B \cup C)\). Since every element of \(C\) is also an element of \(A\), we can also conclude that the intersection and union of set \(A\), \(B\) and \(C\) are equal. This proves the first part of the given statement.
03

Assume \((A \cap B) \cup C=A \cap(B \cup C)\)

Now, we will assume that \((A \cap B) \cup C = A \cap (B \cup C)\). This is the second part of the statement that needs to be proved. The goal is to show that if this equality holds, then it must be that \(C \subseteq A\).
04

Show \(C \subseteq A\)

If we have that \((A \cap B) \cup C = A \cap (B \cup C)\), then it implies that every element of \(C\) that is not in \(A \cap B\), is in \(A\). Hence, all elements of \(C\) are in \(A\). Thus, it must be that \(C \subseteq A\).

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Most popular questions from this chapter

Show the distributive laws $$ A \cap(B \cup C)=(A \cap B) \cup(A \cap C), \quad A \cup(B \cap C)=(A \cup B) \cap(A \cup C) $$

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