What is the Identity for Unions and Intersections of Sets?

  • MHB
  • Thread starter Dustinsfl
  • Start date
In summary, the given identity shows that the intersection of the three sets $(A\cup B), (B\cup C), (C\cup A)$ is equivalent to the union of the pairwise intersections of $A, B, C$. This can be proven by showing that any element in the intersection of the three sets must also be in at least one of the pairwise intersections, and vice versa.
  • #1
Dustinsfl
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$(A\cup B)\cap (B\cup C)\cap (C\cup A) = (A\cap B)\cup (A\cap C)\cup (B\cap C)$

For the identity, we will show $(A\cup B)\cap (B\cup C)\cap (C\cup A) \subseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$ and $(A\cup B)\cap (B\cup C)\cap (C\cup A) \supseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$.
Let $x\in (A\cup B)\cap (B\cup C)\cap (C\cup A)$.
Then $x\in A\cup B$ and $x\in B\cup C$ and $x\in C\cup A$.
So $x\in A$ or $x\in B$ and $x\in B$ or $x\in C$ and $x\in C$ or $x\in A$.

So I am stuck at this point.
 
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  • #2
If ($x \in A$ or $x \in B$) and ($x \in B$ or $x \in C$) and ($x \in C$ or $x \in A$), then ($x \in A$ and $x \in B$) or ($x \in A$ and $x \in C$) or ($x \in B$ and $x \in C$). From there, $x \in (A \cap B) \cup (A \cap C) \cup (B \cap C)$.
 

1. What are unions and intersections?

Unions and intersections are two fundamental concepts in set theory. They refer to the combination or comparison of sets and their elements. A union is the combination of all elements from two or more sets, while an intersection is the common elements shared by two or more sets.

2. How are unions and intersections represented?

Unions are typically represented by the symbol ∪ and intersections by the symbol ∩. For example, A ∪ B represents the union of sets A and B, while A ∩ B represents the intersection of sets A and B.

3. What is the difference between unions and intersections?

The main difference between unions and intersections is that unions combine elements from multiple sets, while intersections only include common elements shared by multiple sets. In other words, unions expand the set, while intersections narrow it down.

4. What are some real-life examples of unions and intersections?

A real-life example of a union could be a library that combines books from two different collections, while an intersection could be a Venn diagram that shows the common interests between two people.

5. How are unions and intersections useful in science?

Unions and intersections are useful in science as they allow us to compare and combine data from different sets, which can help us identify patterns and relationships. They are also used in many mathematical and statistical calculations, such as probability and genetics.

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