Math Problem Statement

',

Let X, Y be two sets. Then X ∪ (X ∩ Y) is?

Solution

The problem is asking to evaluate the expression X(XY)X \cup (X \cap Y) where XX and YY are two sets.

Explanation:

  • XYX \cap Y represents the intersection of sets XX and YY, i.e., the set of elements that are in both XX and YY.
  • X(XY)X \cup (X \cap Y) represents the union of XX with the intersection of XX and YY.

Now, consider the following logic:

  • Every element of XYX \cap Y is already in XX, since the intersection is a subset of XX.
  • Therefore, the union of XX with XYX \cap Y will simply give back the set XX because all elements of XYX \cap Y are already included in XX.

Thus, we conclude: X(XY)=XX \cup (X \cap Y) = X

The correct answer is A\boxed{A}.

Would you like more details on set operations or related concepts?

Here are 5 follow-up questions:

  1. What is the difference between the union and intersection of two sets?
  2. Can you give an example where X(XY)XYX \cup (X \cap Y) \neq X \cup Y?
  3. How would you prove that X(XY)=XX \cup (X \cap Y) = X using set properties?
  4. What happens to the union and intersection if XX and YY are disjoint sets?
  5. Can the union of two sets ever result in a set smaller than the individual sets?

Tip: Remember that the union of two sets ABA \cup B includes all elements from both AA and BB, while the intersection ABA \cap B includes only the common elements.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Set Theory
Union and Intersection of Sets

Formulas

X ∪ (X ∩ Y) = X

Theorems

Subset Theorem
Distributive Law of Sets

Suitable Grade Level

Grades 9-12