Math Problem Statement
5 - Sets Again.pdf
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Solution
The document contains a proof assignment related to sets in mathematics. Here is a breakdown of the problems:
- Problem 1: Suppose , , and are sets. Prove the following:
a.
b.
These are two classic set-theoretic identities. Here’s a sketch of the proofs:
Proof for 1a:
Distributive Property of Intersection over Union
To prove:
Step-by-Step Proof:
-
Start with the left-hand side .
- An element means that and .
- By definition of union, means or .
- Therefore, or , which implies .
-
Now, the reverse direction: Take .
- This means that or , so and .
- Hence, and , which implies .
Since both directions hold, we have proved that .
Proof for 1b:
Distributive Property of Cartesian Product over Intersection
To prove:
Step-by-Step Proof:
-
Start with the left-hand side .
- An element means that and .
- By the definition of intersection, means and .
- Therefore, and , which implies .
-
Now, the reverse direction: Take .
- This means that and , so , , and .
- Hence, .
Since both directions hold, we have proved that .
Would you like further details or clarifications? Here are some related questions to consider:
- What are the laws of set theory, such as De Morgan’s laws, that can help in these types of proofs?
- How do you represent sets graphically to understand the relationships between union, intersection, and Cartesian products?
- Can these distributive properties extend to larger collections of sets (e.g., more than two sets)?
- How do Cartesian products apply in fields like computer science or graph theory?
- What other operations in set theory, like complement or difference, can also be distributed?
Tip: When proving set identities, it's often helpful to work from both sides of the equation and verify that each element follows logically from the assumptions.
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Math Problem Analysis
Mathematical Concepts
Set Theory
Intersection
Union
Cartesian Product
Formulas
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A × (B ∩ C) = (A × B) ∩ (A × C)
Theorems
Distributive Property of Intersection over Union
Distributive Property of Cartesian Product over Intersection
Suitable Grade Level
Undergraduate
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