Exponents and Simplifying Expressions

Exponents and Cartesian Product

Since exponents represent repeated multiplication, exponents may also be used to represent Cartesian products of sets (the set of ordered pairs). See Session 9 (Old Session 9). For example, we may write A × A as A².

Examples:

1. A coin is flipped twice. What are the possible outcomes?
   We may solve this problem as a Cartesian product question by giving the set all the possible ordered pairs. Let C = {H, T}. Then
   C² = C × C = {(H, H), (H, T), (T, H), (T,T)}.
2. If set A = {1, 2}, then A² = A × A = {(1, 1), (1, 2), (2, 1)}.
   
3.  Let B = {0, 1, 2}. Determine the number of elements in B², check the solution by finding B², and then plot B² on a coordinate plane.

The number of elements is (n(B))² = 3² = 9. Reminder: n(B) is the cardinal number for B.  We have

B² = B × B = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)}.

4. Let D be the set of possible outcomes when rolling one 6-sided die. Write D using roster notation. Write out D² using roster notation. What does D² represent? How many elements are in D²?

Solutions: D = {1, 2, 3, 4, 5, 6}

D² = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

D² represents the possible outcomes of rolling a die twice or of rolling two dice.
There are 6² = 36 elements in D².

Self-Check Problem

Write the Cartesian Product for the possible outcomes of a person twice choosing one number between 1 and 3, inclusive.

Solution

return to top | previous page | next page