There are several natural operations for creating new sets out of old ones.

  • the union of two sets \(A\) and \(B\) is the set of all the elements in \(A\) and \(B\) combined
\[A\cup B = \{x: x\in A\ \text{or}\ x\in B\}.\]
  • the intersection of two sets \(A\) and \(B\) is the set of all the elements \(A\) and \(B\) have in common
\[A\cap B = \{x: x\in A\ \text{and}\ x\in B\}.\]
  • the Cartesian product or simply product of two sets \(A\) and \(B\) is the set of all pairs of elements
\[A\times B= \{(a,b): a\in A,\ b\in B\}.\]
  • the complement of a set \(A\) is the set of all the elements in the universe \(U\) not in \(A\)
\[A' = \{x\in U: x\notin A\}.\]

Here \(U\) is the “universe” that we are working inside, whose value should be determined from the context.

Put together, these operations satisfy several natural properties. One such property is distributivity:

\[A\cap(B\cup C) = (A\cap B)\cup (A\cap C)\] \[A\cup(B\cap C) = (A\cup B)\cap (A\cup C)\]

Additionally, taking complements swaps unions and intersections. This is sometimes referred to as De Morgan’s law:

\[(A\cup B)' = A'\cap B'\] \[(A\cap B)' = A'\cup B'\]

Question: Decide if each of the following statements about algebra with sets is TRUE or FALSE.

  • (A) the complement of \(A'\) is \(A\)
  • (B) \(A\cup B=B\) if and only if \(B\subseteq A\)
  • (C) \(A\cap B=B\) if and only if \(B\subseteq A\)
Reveal answer for (A). TRUE. Carefully working through the definition, you should be able to see that taking the complement of a complement gets you back to where you started.
Reveal answer for (B). FALSE. Actually the first statement is equivalent to A being a subset of B
Reveal answer for (C). TRUE. Try drawing a Venn diagram to see that these two conditions are the same.