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 A Δ C = B Δ C ,   then   A = B .

To prove:

If AΔC=BΔC, then A=B.

Given information:

A,B and C be any sets.

Concept used:

AΔB=(A−B)∪(B−A).

Calculation:

The following are the Venn diagrams for AΔC and BΔC :

The shaded portions denote the area covered by the symmetric difference.

AΔC=(A−C)∪(C−A)BΔC=(B−C)∪(C−B)

From the Venn diagram, it can be seen that if an element x∈A

then it is present in either of the two regions:

x∈A−Cx∈A∩C

Analyze both the above cases as follows:

Case 1: x∈A−C

The following two things are observed in this case using the above Venn

diagram:

x∉C Because x∈A−C

x∈AΔC Because the region where x belongs to is contained in the AΔC region.

Since it is given that AΔC=BΔC therefore x∈BΔC.

But x∉C, therefore, x must belong to B.

Case2: x∈A∩C

The following two things are observed in this case using the above Venn diagram:

x∈C because x∈A∩C

x∉AΔC because the region where x belongs to is not contained in the AΔC region.Since, it is given that AΔC=BΔC, therefore x∉(BΔC) either.

But x∈C, therefore, x must belong to B∩C because that is the only region that contains C

But is itself not contained in (BΔC) therefore x∈B.

So, it is concluded that every element that belongs to A also belongs to B.

Thus, A is a subset of B

A⊆B