Chapter 3-1 Relations and Ordered Pairs

Alg. 2 Notes

Cartesian Product

A x B means: The product (x) of sets A and B

is the set of all ordered pairs having the 1st

member from set A and the 2nd from set B.

Example 1:

Find the Cartesian product Q x Q , where Q = {2,3,4}

Answer:

{(2,2), (2,3), (2,4),(3,2), (3,3), (3,4),(4,2), (4,3), (4,4)}

Notice a 3 x 3 Set has 9 elements!!

Example 2

Find the cartesian product A x B where set

A = {stinky, farty}

B = {Carl, James, Cody}

Answer:

{(stinky,Carl),(stinky, James), (stinky, Cody)

(farty,Carl), (farty,James), (farty, Cody)}

Notice: a 2 x 3 product has 6 elements!!

Domain and Range

Domain (x) = the set of all the first members

Range (y) = the set of all the second members

Example 3

List the domain and range of the relation {(a,1), (b,2), (c,3), (e,2)}

Answer:

Domain: {a,b,c,e}Range: {1,2,3,} Notice repeated elements not listed twice

When you list individual elements it is called roster notation

Set Builder Notation:

{x I x 3} read as “the set of all x such that x is less than or equal to 3”

Example 4:Use the set {1,2,3,…,10} to find {x I 4 < x < 7}

Answer:

{5,6}

Example 5

Let Q = {2,3,4} (Ex 1)

Use Q x Q to find {(x,y) x < y}

(read: Use the cartesian product of Q to find members such that the x-value is less than the y-value.

Answer: {(2,3), (2,4), (3,4)}

Example 6: You Try

Use the relation R x R where R = {1,3,5,7}.

Find {(x,y) I y = x + 2}