To prove that if \( A \subseteq B \), then \( A \cap B = A \), we need to show two things:

1. \( A \cap B \subseteq A \) (i.e., every element of \( A \cap B \) is also an element of \( A \)).
2. \( A \subseteq A \cap B \) (i.e., every element of \( A \) is also an element of \( A \cap B \)).

### Proof:

#### 1. Show that \( A \cap B \subseteq A \):
By definition, the intersection of two sets, \( A \cap B \), contains elements that are in both \( A \) and \( B \). So, if \( x \in A \cap B \), then \( x \in A \) and \( x \in B \).

Thus, any element in \( A \cap B \) is clearly an element of \( A \), which means \( A \cap B \subseteq A \).

#### 2. Show that \( A \subseteq A \cap B \):
We are given that \( A \subseteq B \), which means that if \( x \in A \), then \( x \in B \). Therefore, if \( x \in A \), it must also be in \( B \), meaning \( x \in A \cap B \).

Thus, any element in \( A \) is also in \( A \cap B \), which means \( A \subseteq A \cap B \).

### Conclusion:
Since we've shown both \( A \cap B \subseteq A \) and \( A \subseteq A \cap B \), it follows that \( A \cap B = A \).

---

Do you want further details or have any questions? Here are some related problems to explore:

1. Prove that if \( A \subseteq B \), then \( A \cup B = B \).
2. Prove that \( A \cap A = A \).
3. Prove that \( A \cap B \subseteq B \).
4. Prove that \( A \subseteq B \) if and only if \( A \cup B = B \).
5. Prove that \( A \cap \emptyset = \emptyset \).

**Tip:** When proving set equalities, always break down the proof into subset relations to systematically establish both directions of the equality.

Explore the proof for the statement: If A is a subset of B, then A ∩ B = A. This problem involves key concepts from set theory, including the definition of subsets and set intersection. A step-by-step solution is provided for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation