For any two setsAandB(A-B)union(B-A)
Answers
Step-by-step explanation:
If A and B are considered as sets, then
( A - B ) U ( B - A ) gives us what is called as the symmetric difference of the sets A and B. Actually, this is also equal to the quantity (A U B)- (A intersection B)…( sorry I don't have the symbol for intersection :p )
For example, let us take the sets A and B as:-
A = {1,2,3,4,5} and B = {3,4,5,6}
Then, A-B= {1,2} and B-A= {6}
And (A-B) U (B-A) = {1,2,6}
Also,
A U B = {1,2,3,4,5,6} and
A intersection B = {3,4,5}
And (AUB)-(A intersection B) = {1,2,6}, which is same as (A-B) U (B-A)…
So, in order to find (A-B) U (B-A) i.e. the symmetric difference of the sets A and B, it is sufficient to find (AUB)-(A intersection B) .. :-)
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You know that
(A−B)∪(B−A)=A∪B (A−B)∪(B−A)=A∪B
and you want to prove that
A∩B=∅. A∩B=∅.
Say that there is an element
x∈A∩B x∈A∩B . You want to prove that no such exist, so assume that is does. Then Now then
x∈A x∈A and
x∈B x∈B . So, certainly,
a∈A∪B=(A−B)∪(B−A). a∈A∪B=(A−B)∪(B−A). If an element is in the union of two sets, then it is one of the sets (maybe in both). So
x∈A−B x∈A−B or
x∈B−A x∈B−A . But both of these options don't hold. Saying that, for example,
x∈A−B x∈A−B is saying that
x∉B x∉B which contradicts that
x∈A∩B x∈A∩B . Hence no such
Essentially you mean the set that contains those elements that belong to set A and not in set B , as well as those elements that belong to set B and not in set A .
So that means it includes all elements in A or B minus those that are common to both A and B .
Therefore,
(A−B)∪(B−A)=(A∪B)−(A∩B)