Refer to the definition of symmetric difference given above. Prove each of 47-52, assuming that A, B, and C are all subsets of a universal set U.
A,B and C be any sets.
The following are the Venn diagrams for :
The shaded portions denote the area covered by the symmetric difference.
From the Venn diagram, it can be seen that if an element
then it is present in either of the two regions:
Analyze both the above cases as follows:
The following two things are observed in this case using the above Venn
Because the region where belongs to is contained in the region.
Since it is given that therefore .
But , therefore, must belong to .
The following two things are observed in this case using the above Venn diagram:
But , therefore, must belong to because that is the only region that contains
But is itself not contained in therefore .
So, it is concluded that every element that belongs to also belongs to .
Thus, is a subset of
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