Sets:Operations

Definitions

The two main set operations that we deal with are union and intersection. The union of two sets $A$ and $B$ is defined as $A\cup B=\{x:x\in A$ or $x\in B\}$. For example:

• The union of $A=\{1,3,5,7,9\}$ and $B=\{0,1,2,4,6,8,9\}$ is $A\cup B=\{0,1,2,3,4,5,6,7,8,9\}$
• The union of the even integers and odd integers is $\mathbb {Z}$.
• The union of the set of rational numbers and the set of irrational numbers is $\mathbb {R}$.
• $A\cup A=A$, and $\emptyset \cup A=A$.
• For sets $A$ and $B$ such that $A\subseteq B$, $A\cup B=B$, since all elements of $A$ are already in $B$ if $A\subseteq B$.

The intersection of $A$ and $B$ is defined as $A\cap B=\{x:x\in A$ and $x\in B\}$; that is, the intersection of $A$ and $B$ is the set of all elements shared by the two sets. Sets $A$ and $B$ are "disjoint" if $A\cap B=\emptyset$.

• The intersection of $A=\{1,3,5,7,9\}$ and $B=\{0,1,2,4,6,8,9\}$ is $A\cap B=\{1,9\}$.
• The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd).
• $A\cap A=A$, and $\emptyset \cap A=\emptyset$.
• For sets $A$ and $B$ such that $A\subseteq B$, $A\cup B=A$.

There are a few other common set operations. The set difference of $A$ and $B$ is defined as $A\backslash B=\{x:x\in A,x\not \in B\}$. We read $A\backslash B$ (also sometimes denoted as $A-B$) as "$A$ without $B$". Note that this operation is not commutative; that is, $A\backslash B$ does not equal $B\backslash A$ in most cases. Example: if $A=\{1,2,3,4,5\}$ and $B=\{0,1,3,4,5,6\}$, then $A\backslash B=\{2\}$ and $B\backslash A=\{0,6\}$.

Another set operation is the Cartesian product (or the product of two sets). The Cartesian product of two sets $A$ and $B$ is defined as $A\times B=\{(a,b):a\in A$ and $b\in B\}$, where $(a,b)$ is an ordered pair. For example, if $A=\{1,2,3\}$ and $B=\{0,1\}$, then $A\times B=\{(1,0),(1,1),(2,0),(2,1),(3,0),(3,1)\}$. If $A$ and $B$ are both finite sets, then the number of elements in $A\times B$ is the product of the number of elements in set $A$ and set $B$ (for example, if $A$ has 3 elements and $B$ has 2, then $A\times B$ has 6).

Resources

• Course Textbook, pages 101-115