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 (AB)(BA)(A - B) \cup (B - A).

Step 1: Simplifying (AB)(BA)(A - B) \cup (B - A)

  • ABA - B means all the elements that are in AA but not in BB.
  • BAB - A means all the elements that are in BB but not in AA.

Thus, (AB)(BA)(A - B) \cup (B - A) represents all the elements that are in AA or in BB, but not in both. This is the symmetric difference of AA and BB, often denoted as AΔBA \Delta B.

(AB)(BA)=AΔB(A - B) \cup (B - A) = A \Delta B

Step 2: Analyze each option

Option 1: (AB)(AB)(A \cap B)' \cap (A \cup B)

  • (AB)(A \cap B)' means the complement of ABA \cap B, i.e., all elements not in both AA and BB.
  • (AB)(A \cup B) means all elements that are in AA or in BB or in both.

The intersection of (AB)(A \cap B)' with (AB)(A \cup B) represents the elements in ABA \cup B that are not in both sets AA and BB. This is exactly the definition of AΔBA \Delta B.

Thus, this option simplifies to AΔBA \Delta B, which matches (AB)(BA)(A - B) \cup (B - A).

Option 2: (AB)(AB)(A \cap B) \cap (A \cup B)'

  • (AB)(A \cup B)' is the complement of ABA \cup B, i.e., all elements that are in neither AA nor BB.
  • ABA \cap B is the set of elements common to both AA and BB.

The intersection of ABA \cap B with (AB)(A \cup B)' will always be the empty set because there are no elements that are both in ABA \cap B and outside ABA \cup B. Thus, this expression is the empty set, which is not equal to AΔBA \Delta B.

Option 3: (AB)(AB)(A \cup B)' \cup (A \cap B)

  • (AB)(A \cup B)' is the complement of ABA \cup B, i.e., the set of elements not in ABA \cup B.
  • ABA \cap B is the set of elements common to both AA and BB.

The union of (AB)(A \cup B)' and ABA \cap B represents elements that are either outside ABA \cup B or inside both sets AA and BB. This is not equivalent to the symmetric difference AΔBA \Delta B.

Option 4: (AB)(AB)(A \cup B) \cup (A \cap B)'

  • (AB)(A \cap B)' is the complement of ABA \cap B, i.e., all elements not in both AA and BB.
  • (AB)(A \cup B) is the union of AA and BB.

This expression includes all elements in ABA \cup B and all elements not in ABA \cap B, which would result in the universal set UU (everything), as it would contain all possible elements in or out of AA and BB. 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