Just as with numbers, there are a number of ways we can combine or otherwise modify sets.

Summary

Given two sets \(A\) and \(B\),

• \(A\cup B\), the union of the sets, is the set you get by merging the elements of the two
• \(A \cap B\), the intersection of the two, is set of elements common to both
• \(A-B\), the difference of the sets, is the set of elements of \(A\) that aren't also in \(B\).
• \(2^A\), the powerset of \(A\), is the set of subsets of \(A\).  So the powerset of \(\{a,b\}\) is the set of its four possible subsets: \(\{\{\}, \{a\}, \{b\}, \{a,b\}\}\).
• \(A\times B\), the Cartesian product of the sets is the set of all possible pairs formed from an element of \(A\) followed by an element of \(B\).  Will talk about tuples next.

Union \(\cup\)

The union of two sets \(A\) and \(B\) is just the set containing all the elements of both:

Since sets can only have an element or not (they can't have an element twice), if an element is in both sets, it's only contained in the union once.

Intersection \(\cap\)

The intersection of two sets \(A\) and \(B\) is just the set of elements common to both:

If the sets are disjoint, then \(A \cap B = \emptyset\).

Difference of two sets

The set of elements in \(A\) but not \(B\) is written \(A-B\):

Powerset

We often want to think about subsets of another set (sets contained in the original set).  The set of all possible subsets of a set \(A\) is called the powerset of \(A\) and is written \(2^A\):

To understand why we notate it \(2^A\), consider the following.  The empty set \(\{\}\) has zero elements and one subset, itself (remember that subset includes equality).  A set with one element, e.g. \(\{ 1 \}\), has two subsets: itself and the empty set.  A set with two elements, e.g. \(\{ 1, 2 \}\) has four possible subsets: \(\{\}\), \(\{ 1\}\), \(\{ 2\}\), and \(\{1,2\}\).  In general, when choosing subsets, we choose whether or not to include each element.  For a set with \(n\) elements, there are \(n\) such choices, with 2 options per choice (include it or not) and so the total number of combinations is \(2^n\). So we use the notation \(2^A\) to express the set of subsets of \(A\) because the number of such sets is 2 to the number of elements in the set.

Important: \(2^A\) is different from \(A^2\), defined below.

Cartesian product: \(\times\)

We haven't talked about pairs and tuples yet, so don't worry if you don't understand this yet. But tuples are simple: they're just lists.  We're including it here so it's together with the other operations. The Cartesian¹ product makes sets of from other sets.  \(A\times B\) is the set of all pairs of elements drawn from \(A\) and $$B$, respectively:

More products make longer tuples: \(A\times B \times C\) is the set of all triples formed from elements of \(A\), \(B\), and \(C\), respectively:

More generally, the product of \(n\) sets is an \(n\)-tuple (list of \(n\) elements):

Exponentiation

The power of a set is defined analogously to powers of numbers: \(S^2 = S\times S\), \(S^3 = S \times S \times S\), and so on.  More generally, \(S^n\) is the set of all \(n\)-tuples (lists of \(n\) elements) whose elements are all drawn from \(S\).

Important \(S^2\), the set of pairs of elements drawn from \(S\), is unrelated to \(2^S\), the set of all subsets of \(S\), other than both involving \(S\).

Cardinality (size of a set)

The number of elements in a set \(S\) is just written \(|S|\).  For the moment, we will assume \(S\) has a finite number of elements.  The esoterica includes a discussion of how to think about the sizes of infinite sets, which turns out to be shockingly hard and counter-intuitive.

Remembering the symbols

It's more important to understand the ideas than the notation.  That said, here are some mnemonics for remembering the notation:

• \(A\cup B\) is a union because it has a little “u”
• \(A \cap B\) is an intersection because it has an upside-down “u” and intersection is a kind of opposite of union
• \(A-B\) is subtracting the elements of $$B$ from \(A\)
• For the multiplication-like operations
  • The powerset of a \(A\) with \(a\) elements is \(2^A\) because has \(2^a\) elements (there are \(2^a\) possible subsets of $$A$)
  • The set of tuples formed from a set \(A\) with \(a\) elements and a set \(B\) with \(b\) elements is notated \(A\times B\) because it has \(a\times b\) elements.
  • The set of \(n\)-tuples formed from the elements of a set \(A\) with \(a\) elements is \(A^n\) both because it is \(A\) multiplied by itself repeatedly, and because it has \(a^n\) elements.

Footnotes