Let A B = (A \\B) U (B \\ A) stand for the symmetricdifference of the sets A and
ID: 3616246 • Letter: L
Question
Let A B = (A B) U (B A) stand for the symmetricdifference of the sets A andB. (U = union) Where Aand B are languages, i.e. subsets of* for some alphabet , it is understoodthat x refers to strings over.) Under what conditionsdoes a) A B = { }b) A B =* c) A B =A Let A B = (A B) U (B A) stand for the symmetricdifference of the sets A andB. (U = union) Where Aand B are languages, i.e. subsets of* for some alphabet , it is understoodthat x refers to strings over.) Under what conditionsdoes a) A B = { }
b) A B =* c) A B =A c) A B =A
Explanation / Answer
a) A B = { } whenA=B The reason is when A=B, A-B ={}, B-A ={ } Therefore A B = (A -B) U (B-A) = { } U { } = { } B) A B = * when AUB =* and A ^ B = { } ...{^ stands for intersectionhere} When the above conditions are satisfied, A-B =A, B-A = B Therefore A B = (A-B) U (B-A) = A U B = * c) A B = A when B={} To prove this we will consider the contradiction that B is notequal to { }..i.e. B has some elements There are two cases in this i) B is a subset of A, in this case (A- B) isa subset of A but not equal to A and B-A = { } ...since B isa subset of A therefore A B =C where C is a subset of A but not equal to A Hence if B is a subsetof A, then A B cannot equals A ii) B is not a subset of A In this case B is not {} and alsonot a subset of A Therefore A-B = C where C is notequal to A B-A = D [ D containssome elements only in B but not in A] A B = C U D which is not equal to A because of D which has someelements which are only in B but not in A Therefore from the above two cases we can say that for A B=A, B should be {}
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