It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

# Set Theory

## Set Operations

The intersection of sets A and B, written $$A \cap B$$ is the set of elements common to both $$A$$ and $$B$$, represented as

$A \cap B = \{x|x\in A \,and\,x\in B\}$

Example: Find the intersection of the following sets, $$A = \{1, 2, 3, 5, 8 \}, \,B=\{1, 2, 3, 4 \}$$

Solution: $$A \cap B =\{1, 2, 3 \}$$ Elements that are in both sets A and B are in the intersection.

The union of sets and B, written $$A \cup B$$, is the set of elements belonging to either of the sets, represented as:

$A \cup B = \{x|x\in A \,or\,x\in B\}$

Example: Find the union of the following sets, $$A = \{1, 2, 3, 5, 8 \}, \,B=\{1, 2, 3, 4 \}$$

Solution: $$A \cup B =\{1, 2, 3, 4, 5, 8 \}$$ All elements that in sets A and B are in the union.

The difference of sets A and B, represented $$A - B$$, is the set of elements belonging to set A and not to set B,

$A - B = \{x|x\in A \,and\,x\notin B\}$

Example: Find the difference of the sets, $$A = \{1, 2, 3, 5, 8 \}, \,B=\{1, 2, 3, 4 \}$$, $$A - B$$, and $$B - A$$

Solution: $$A - B =\{5, 8 \}$$ All elements that are in set A but not in set B.

$$B - A =\{4\}$$ All elements that are in set B but not in set A.

## Ordered Pairs

In the ordered pair $$\left(a,b\right)$$, a is called the first component and b is called the second component. Switching components does not give you the same statement. $$\left(a,b\right)\ \neq \left(b,a\right)$$.

The Cartesian product of sets A and B, written $$A \times B$$, is: $A \times B = \{(a, b)|a\in A \,and\, b\in B\}$

Example: Let $$A = \{1, 2 \}, \, and \,B=\{a, b, c \}$$

Then $$A \times B = \{(1,a), (1,b), (1,c), (2,a), (2,b), (2,c) \}$$

The cardinal number of a Cartesian product is the product of the cardinal numbers of each set.

If $$n\left(A\right)=a$$ and $$n\left(B\right)=b$$, then

\begin{align} n\left(A \times B\right) &= n\left(B \times A\right) \\ &= n\left(A \right) \cdot n\left(B \right) = n\left(B \right) \cdot n\left(A \right) \\ &= ab = ba \end{align}

Example: Let $$A = \{1, 2 \}, \, and \,B=\{a, b, c \}$$, then $$n(A) = 2$$ and $$n(B) = 3$$. $$n(A \times B) = 6$$