Question

Prove $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.

   Prove $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.

Abstract Algebra: Theory and Applications

Abstract Algebra: Theory and Applications

Thomas W. Judson 1st Edition

Chapter 1, Problem 7

Instant Answer

Solved by Expert Khushbu Rani

03/19/2024

Step 1

By the definition of intersection, this means we are looking for all elements \( x \) such that \( x \in A \) and \( x \in (B \cup C) \). Show more…

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Prove $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.

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Key Concepts

Set Intersection

Set intersection refers to the collection of elements that are common to both sets in question. It is a fundamental operation in set theory that helps in identifying elements shared among sets, and it is typically denoted by the symbol ?.

Set Union

Set union represents the combination of all elements present in either of the two sets. This operation, denoted by the symbol ?, forms a set containing all unique elements from both sets, without duplication.

Distributive Property in Set Theory

The distributive property in set theory is a rule that relates the union and intersection operations. Specifically, it states that the intersection of a set with the union of two other sets is equivalent to the union of the intersections of the first set with each of the other sets. This property is essential for simplifying complex set expressions and is analogous to distribution in algebra.

Proof by Double Inclusion

Proof by double inclusion is a common method for proving the equality of two sets. It involves showing that every element of the first set is also an element of the second set (first inclusion), and vice versa (second inclusion). This technique rigorously establishes set equality by confirming that both sets have precisely the same elements.

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