Math Problem Statement
Let A,BA,B be sets. Then (A−B)∪(B−A)(A−B)∪(B−A) is equal to:
(A∩B)′∩(A∪B)(A∩B)′∩(A∪B)
(A∩B)∩(A∪B)′(A∩B)∩(A∪B)′
(A∪B)′∪(A∩B)(A∪B)′∪(A∩B)
(A∪B)∪(A∩B)′(A∪B)∪(A∩B)′
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Solution
We are asked to find which of the given expressions is equal to .
Step 1: Simplifying
- means all the elements that are in but not in .
- means all the elements that are in but not in .
Thus, represents all the elements that are in or in , but not in both. This is the symmetric difference of and , often denoted as .
Step 2: Analyze each option
Option 1:
- means the complement of , i.e., all elements not in both and .
- means all elements that are in or in or in both.
The intersection of with represents the elements in that are not in both sets and . This is exactly the definition of .
Thus, this option simplifies to , which matches .
Option 2:
- is the complement of , i.e., all elements that are in neither nor .
- is the set of elements common to both and .
The intersection of with will always be the empty set because there are no elements that are both in and outside . Thus, this expression is the empty set, which is not equal to .
Option 3:
- is the complement of , i.e., the set of elements not in .
- is the set of elements common to both and .
The union of and represents elements that are either outside or inside both sets and . This is not equivalent to the symmetric difference .
Option 4:
- is the complement of , i.e., all elements not in both and .
- is the union of and .
This expression includes all elements in and all elements not in , which would result in the universal set (everything), as it would contain all possible elements in or out of and . Thus, this
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Math Problem Analysis
Mathematical Concepts
Set Theory
Symmetric Difference
Set Operations
Complement of Sets
Union and Intersection
Formulas
(A - B) ∪ (B - A) = A Δ B
De Morgan's Law: (A ∩ B)' = A' ∪ B'
Theorems
De Morgan's Laws
Properties of Set Operations
Symmetric Difference Definition
Suitable Grade Level
Grades 10-12