Simplify the view of the set

Question

Simplify the view of the set

To simplify the form of a set given by operations, applying the laws of set algebra (the answer sets can enter no more than once): (A ∩ B)∆(¯A ∩ B) What's next? Can I take B and move to left B?

1

дискретная-математика множества

Author: Bohdan, 2020-09-30

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1 answers

score 1 · Accepted Answer

In general, it is convenient to use drawings here. You draw two intersecting circles: set A and set B (there are drawings, for example, here). Next, you use a different color for two sets: (A ∩ B) and (A ∩ B). Everything that is colored with the 1st and 2nd colors, but not both, is the answer. I think it will be easy to understand that the answer is A∆B.

If we go the formal way, then

(A ∩ ¯B)∆(¯A ∩ B) = ((A ∩ ¯B) ⋃ (¯A ∩ B)) \ ((A ∩ ¯B) ⋂ (¯A ∩ B)) = A∆B \ ∅ = A∆B, т.к.:
(A ∩ ¯B) ⋃ (¯A ∩ B) = {x: (x∈A & x∉B) ∨ (x∉A & x∈B)} = A∆B
(A ∩ ¯B) ⋂ (¯A ∩ B) = {x: (x∈A & x∉B) & (x∉A & x∈B)} = ∅