I am trying to prove this as false. Let A, B, C be any three sets. If A U C = B U C then B = C. I can draw a Venn Diagram to prove this and I can assign values to the sets to prove it, but how can I prove without doing this? Also is the counter value A U C = B U C then B not equal to C? Can you prove using this? I am a bit confused by this problem. Maybe I am thinking too hard about it.