Math Problem Statement
Prove A ∪ B = B ∪ A and A ∩ B = B ∩ A
Solution
We are tasked with proving the following two set identities:
1. Proving
To prove that , we need to show that every element in is also in , and vice versa.
Step 1: Prove
- Let . By definition of the union of sets, this means or .
- If , then (since is in , and includes all elements of ).
- If , then (since is in , and includes all elements of ).
- Therefore, , proving that .
Step 2: Prove
- Let . By definition of the union of sets, this means or .
- If , then (since is in , and includes all elements of ).
- If , then (since is in , and includes all elements of ).
- Therefore, , proving that .
Since both inclusions hold, we have shown that .
2. Proving
Next, we need to prove that . Again, we show that every element of is in , and vice versa.
Step 1: Prove
- Let . By the definition of intersection, this means that and .
- Since and , it follows that (because includes all elements that are in both and ).
- Therefore, , proving that .
Step 2: Prove
- Let . By the definition of intersection, this means that and .
- Since and , it follows that (because includes all elements that are in both and ).
- Therefore, , proving that .
Since both inclusions hold, we have shown that .
Conclusion
We have proven both identities:
Would you like more details or further clarification on any part of the proof?
Here are 5 related questions:
- What are the properties of set unions and intersections in general?
- How can De Morgan's Laws be applied to set operations?
- What is the difference between the symmetric difference and union/intersection of sets?
- How can Venn diagrams help visualize set identities?
- How do set operations behave in the context of finite and infinite sets?
Tip: When proving set identities, it's helpful to break the problem into smaller inclusions (showing both subsets) to demonstrate equality.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Union of Sets
Intersection of Sets
Set Identities
Formulas
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Theorems
Commutative Property of Set Union
Commutative Property of Set Intersection
Suitable Grade Level
Grades 8-10