Can you conclude that A = B if A, B, and C are sets such that
A ∪ C = B ∪ C, A ∩ C = B ∩ C, A ∪ C = B ∪ C , A ∩ C = B ∩ C

Solution:

If A ∪ C = B ∪ C it necessarily means that A = B. Unless all the elements of A and B are the same their union with a third set C cannot be the same.

If A ∩ C = B ∩ C does not necessarily imply that A = B because all elements of A and B may not be same. Only elements common with C may be same. The rest of elements contained in A and B may not be same as shown below:

A = {1, 7, 8, 9, 11} C = {2, 4,8, 9} A ∩ C = {8, 9}

B = {8, 9, 13, 14} C = {2, 4,, 8, 9} B ∩ C = {8, 9}

It is apparent from the above two venn diagrams that even though

A ∩ C = B ∩ C = {8.9}

A ≠ B

A = {1, 7, 8, 9, 11}

B = {8, 9, 13, 14}

Can you conclude that A = B if A, B, and C are sets such that
A ∪ C = B ∪ C, A ∩ C = B ∩ C, A ∪ C = B ∪ C , A ∩ C = B ∩ C

Summary:

The problem statement is whether it is possible to conclude that A = B if A, B, and C are sets such that: 1)A ∪ C = B ∪ C 2)A ∩ C = B ∩ C. The conclusion drawn is that it is possible to infer A = B only if A ∪ C = B ∪ C.