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Symmetric difference
Task number: 3343
Determine which of the following statements about the symmetric difference ⊕ defined as A⊕B=(A∪B)∖(A∩B) are true, and which are false.
- A⊕B=(A∩¯B)∪(¯A∩B)
- A⊕B=B⊕A
- A⊕(B⊕C)=(A⊕B)⊕C
- A⊕(B⊕A)=A
- A⊕A=∅
- A⊕∅=A
If a statement is false, correct it using the smallest possible change if possible.
Solution
For the first, third and fourth statements it's helpful to draw a Venn diagram and mark the sets corresponding to the expressions on the left and right.
It's possible to prove some of the statements formally, e.g. by transforming set expressions, e.g.
A⊕B=(A∪B)∖(A∩B)=(B∪A)∖(B∩A)=B⊕A. Here we have used the fact that ∩ and ∪ are commutative.
A⊕A=(A∪A)∖(A∩A)=A∖A=∅
A⊕∅=(A∪∅)∖(A∩∅)=A∖∅=A
We can refute the fourth statement formally using the second, third and fifth:
A⊕(B⊕A)=A⊕(A⊕B)=(A⊕A)⊕B=∅⊕B=B⊕∅=BAnswer
A⊕(B⊕A)=A is invalid; the statement A⊕(B⊕A)=B e.g. is true.