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.
If
To prove:
If
Given information:
A,B and C be any sets.
Concept used:
Calculation:
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:
Case 1:
The following two things are observed in this case using the above Venn
diagram:
Since it is given that
But
Case2:
The following two things are observed in this case using the above Venn diagram:
But
But is itself not contained in
So, it is concluded that every element that belongs to
Thus,