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The most direct way (not always the simplest) to prove that "X=Y", for sets, is to prove both X⊆Y and Y⊆X. And to prove "X⊆Y" start with "if x∈X" then use the properties of sets X and Y to conclude "then x∈Y".
Here the two sets are X=(A∪B)−(A∩B) and Y=(A−B)∪(B−A). If x∈(A∪B)−(A∩B) then x∈A∪B) and x∉A∩B. That means x is in either A or B but not both.
Case 1: Suppose x is in A but not in B. Then x is in A−B so in (A−B)∪(B−A). Case 2: Suppose x is in B but not in A. Then x is in B−A so in (A−B)∪(B−A). In either case, (A∪B)−(A∩B)⊆(A−B)∪(B−A).
Now the other way- Suppose x∈(A−B)∪(B−A). Then either Case 1: x∈A−B. Then x∈A∪B but not in A∩B so x∈(A∪B)−(A∩B). Case 2: x∈B=A. Then, again, x∈A∪B but not in A∩B so x∈(A∪B)−(A∩B). In either case, ∪(B−A)⊆(A∪B)−(A∩B).
Therefore, (A∪B)−(A∩B)=(A−B)∪(B−A).
This implies A = B.
<h3>What is set theory?</h3>
The area of mathematical logic known as set theory examines sets, which are essentially collections of objects. Set theory is an area of mathematics that primarily deals with objects that are pertinent to mathematics as a whole, despite the fact that any form of object can be gathered into a set.
To learn more about set theory refer to:
brainly.com/question/16809468
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