I am beginning to learn set theory proofs. It has been extremely useful to draw Venn diagrams for proofs just involving union, intersection, complement, e.t.c. However with cross product involved, how can I gain intuition regarding if a statement is true or false?
If the problem states to prove it I can do it but have trouble if I have to determine the initial truth value.
Here is an example of what I mean:
Question: Consider the following two statements about sets A, B, C, D. One is true, and the other is false.
- (A − B) × (C − D) ⊆ (A × C) − (B × D)
- (A × C) − (B × D) ⊆ (A − B) × (C − D)
(a) Write the number of the true statement, and write a proof of the statement.
(b) Write the number of the false statement, and write a specific counterexample where A, B, C, D are subsets of the set Z of integers.
Let $A=(1,3)$ and $B=(2,4)$ be intervals along the $x$-axis, and $C$ and $D$ be the same intervals, but along the $y$-axis. These will break up their axes into 5 regions (don't sweat whether or not they are closed or open intervals - it won't matter in the end).
Now when you take all the cross products they will split the $x$-$y$ plane into $5\times 5 =25$ regions. You can then shade in the areas for each combination given, and see which lie inside which.
For your part B, you can adjust the values so that each region actually contains some integers.