What operations can be used to combine sets?

"Okay, so we have sets. Now what can we do with them? When you first learn about numbers back before kindergarten, the next thing you learn is how to combine numbers using various operations to produce other numbers. These include + , − , × , ÷ , exponents, roots, etc. Sets, too, have operations that are useful for combining to make other sets. These include: • Union ( ∪ ). The union of two sets is a set that includes the elements that either (or both) of them have as members. For instance, if A = { Dad, Lizzy }, and B = { Lizzy, T.J., Marina }, then A ∪ B = { Dad, Lizzy, T.J., Marina }. Note that an element is in the union if it is in A or B . For this reason, there is a strong relationship between the union operator of sets and the “or” ( ∨ ) operator of boolean logic that we’ll see later. • Intersection ( ∩ ). The intersection of two sets is a set that includes the elements that both of them have as members. In the above example, A ∩ B = { Lizzy }. There is a strong connection between intersection and the “and” ( ∧ ) boolean logic operator. • (Partial) complement ( − ). Looks like subtraction, but significantly different. A − B contains the elements from A that are not also in B . So you start with A , and then “subtract off” the contents of B , if they occur. In the above example, A − B = { Dad }. (Note that T.J. and Marina didn’t really enter in to the calculation.) Unlike ∪ and ∩ , − is not commutative . This means it’s not symmetrical: A − B doesn’t (normally) give the same answer as B − A . In this example, B − A is { T.J., Marina }, whereas if you ever reverse the operands with union or intersection, you’ll always get the same result as before. • (Total) complement ( X ). Same as the partial complement, above, except that the implied first operand is Ω . In other words, A − B is “all the things in A that aren’t in B ,” whereas B is “all the things period that aren’t in B .” Of course, “all the things period” means “all the things that we’re currently talking about.” The domain of discourse Ω is very important here. If we’re talking about the Davies family, we would say that M = { Mom, Lizzy }, because those are all the Davieses who aren’t male. If, on the other hand, Ω is “the grand set of absolutely everything,” then not only is Mom a member of M , but so is the number 12, the French Revolution, and my nightmare last Tuesday about a rabid platypus. • Cartesian product ( × ). Looks like multiplication, but very different. When you take the Cartesian product of two sets A and B , you don’t even get the elements from the sets in the result. Instead, you get ordered pairs of elements. These ordered pairs represent each combination of an element from A and an element from B. For instance, suppose A = { Bob, Dave } and B = { Jenny, Gabrielle, and Tiffany }. Then: A × B = { (Bob, Jenny), (Bob, Gabrielle), (Bob, Tiffany), (Dave, Jenny), (Dave, Gabrielle), (Dave, Tiffany) }. Study that list. The first thing to realize is that it consists of neither guys nor girls, but of ordered pairs. (Clearly, for example, Jenny / ∈ A × B .) Every guy appears exactly once with every girl, and the guy is always the first element of the ordered pair. Since we have two guys and three girls, there are six elements in the result, which is an easy way to remember the × sign that represents Cartesian product. (Do not, however, make the common mistake of thinking that A × B is 6. A × B is a set, not a number. The cardinality of the set, of course, is 6, so it’s appropriate to write | A × B | = 6 .)"