Show that if A ⊆ B,
then A ∪ B = B.show all the steps

We are tasked with proving that if \( A \subseteq B \), then \( A \cup B = B \). Let's break it down step by step.

### Definitions:

- \( A \subseteq B \) means that every element of \( A \) is also an element of \( B \).
- \( A \cup B \) denotes the union of sets \( A \) and \( B \), which contains all elements that are in \( A \), \( B \), or both.

### Proof:

We need to show that \( A \cup B = B \) under the assumption \( A \subseteq B \).

#### Step 1: Show that \( A \cup B \subseteq B \)

- By the definition of union, \( A \cup B \) contains elements that are in \( A \), \( B \), or both.
- But we know \( A \subseteq B \), meaning every element of \( A \) is already in \( B \).
- Thus, all elements of \( A \cup B \) are in \( B \), since any element in \( A \) is already in \( B \), and any element in \( B \) is obviously in \( B \).
- Therefore, \( A \cup B \subseteq B \).

#### Step 2: Show that \( B \subseteq A \cup B \)

- By the definition of union, \( A \cup B \) contains all elements of \( B \), because \( A \cup B \) includes any element that is in \( B \).
- Therefore, every element of \( B \) is in \( A \cup B \).
- Thus, \( B \subseteq A \cup B \).

#### Step 3: Conclude that \( A \cup B = B \)

- From Step 1, we have \( A \cup B \subseteq B \).
- From Step 2, we have \( B \subseteq A \cup B \).
- Since both inclusions hold, we conclude that \( A \cup B = B \).

### Final Conclusion:
If \( A \subseteq B \), then \( A \cup B = B \).

Would you like more details or have any questions?

Here are 5 questions to consider:
1. What is the difference between \( A \subseteq B \) and \( A \subset B \)?
2. Can you think of an example where \( A \cup B = B \), even if \( A \neq B \)?
3. How does the union operation behave with disjoint sets?
4. What happens to the union when \( A \cap B = \emptyset \)?
5. Can \( A \cup B = A \) ever be true if \( A \subseteq B \)?

**Tip:** When proving set equalities, it is often useful to show mutual inclusion — that each set is a subset of the other.

Show that if A ⊆ B, then A ∪ B = B. Show all the steps.

This problem demonstrates how to prove that if A is a subset of B (A ⊆ B), then the union of A and B (A ∪ B) equals B. The solution covers key set theory concepts, including the definitions of subsets and union. Suitable 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