Math Problem Statement
- Give a general prove of the statement below. Let X and Y be two non-empty sets. If Y ⊆ X, then show that X ∪ Y = X.
Solution
To prove the statement "If , then " for any two non-empty sets and , we will use the definitions of set union and subset.
Definitions:
- Union of Sets: The union of two sets and , denoted , is the set of all elements that are in , in , or in both.
- Subset: A set is a subset of a set , denoted , if every element of is also an element of .
Proof:
To prove that when , we need to show two inclusions:
1. Prove :
Let be an arbitrary element of . By the definition of union: We need to show that .
- If , then .
- If , since , .
In both cases, we have . Therefore, Hence, .
2. Prove :
Let be an arbitrary element of . By the definition of union: because satisfies the condition for being in the union .
Thus, Hence, .
Conclusion:
Since we have shown both and , it follows that:
This completes the proof.
Would you like more details on any part of this proof or have any questions?
Here are some related questions you might find interesting:
- How do you prove that if ?
- What is an example where and ?
- How do you define the intersection of two sets?
- Can the empty set be a subset of any set? Why?
- How would you prove that the union of a set with the empty set is the set itself?
- What is the symmetric difference of two sets and how is it defined?
- How do you prove that ?
- What are the properties of set union and intersection that are used in proofs?
Tip: When working with sets, always start by carefully defining the sets and the operations you are using. This helps in constructing clear and rigorous proofs.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Formulas
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Theorems
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Suitable Grade Level
Advanced