Set Relations, Ordered Pairs

A correspondence between two sets is a relation which connects the two sets.

For instance, the set A = {1, 2, 3, 4} and the set B = {1, 8, 27, 64} are related through the statement b = a 3 since each member of set B is the cube of the corresponding element of set A.

Note: Set names are in caps, set elements are represented by the same letter as the parent set in lower case.

Correspondences are also known as mappings.

There are 3 ways to indicate a correspondence, mapping or relation between sets.

They are: a correspondence table, an algebraic statement of the relation, or a set of ordered pairs.

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b = a3 is an algebraic statement of the same relation or correspondence shown in the table.

C = {(1, 1) (2, 8) (3, 27) (4, 64)} is the set of ordered pairs which represents exactly the same relation or correspondence as shown in the table and equation above.

Example 2: If A = {1, 3, 5, 7}, write an equation for and list the ordered pairs in the

set B whose members are the squares of the elements in A.

Solution: the equation is b = a2 and the ordered pairs are {(1, 1) (3, 9) (5, 25) (7, 49)}.

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 The Cartesian product of two sets A and B, denoted A × B,is the set of all ordered pairswith first element from set A and second element from set B.

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So, if set A = {0, 1} and set B = { – 2, 3, 5, 8}
then A × B = {(0, – 2) (0, 3) (0, 5) (0, 8) (1, – 2) (1, 3) (1, 5) (1, 8)}

As we see there are 8 members in the set A × B

since there are 2 elements in A and
4 elements in B.

Therefore, if set A contains m elements and set B contains n elements,
there will be mn elements in A × B.

Note also that each element of set A is paired with each element of set B.

The order of the entries in the pairs corresponds to the order of the set names,

so elements in A × B are listed as (a, b) and

elements in B × A are listed as (b , a)

Example 3

Find the Cartesian products A × B and B × A if A = {1, 2, 3} and B = {a, b}.

Solution

A × B = {(1, a) (1, b) (2, a) (2, b) (3, a) (3, b)}

B × A = {(a, 1) (a, 2) (a, 3) (b, 1) (b, 2) (b, 3)}

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When we define or select a specific set of ordered pairs from a Cartesian product, we have a relation.

 A relation from a set A to a set B is any set of ordered pairs in A × B which satisfies the relation statement.

If we have the Cartesian product
A × B = {(1, 0) (1, 1) (2, 0) (2, 1) (2, 2) (3, 0) (3, 1) (3, 2) (3, 3)} and we define the equality relation on this set we would select the elements (1,1), (2,2) and (3,3) since these are the members of A × B in which a = b.

The Cartesian plane defined by the X and Y axes is a graphic representation of R × R.

The 1st quadrant of the Cartesian plane is a graphic representation of R + × R + since it includes every positive Real number between zero and infinity on both the x and y axes.

Say we use A × B from above, but now we want to list the elements in the less than relation.

That is, we want all the ordered pairs in which b < a. In this case our set includes the ordered pairs {(1, 0) (2, 0) (2, 1) (3, 0) (3, 1) (3, 2)} since each 2nd element is less than the first in each of these ordered pair.

Had we been asked to find the members of the set in which b > a, the answer would be the null set , since there are no ordered pairs in A × B in which the second element is greater than the first.

Example 4

A × B = {(1, 0) (1, 1) (2, 0) (2, 1) (2, 2) (3, 0) (3, 1) (3, 2) (3, 3)}.

List the ordered pairs in the set in which b = a – 1,

Solution

We need all the ordered pairs in which the 2nd element is 1 less than the first.

The ordered pairs in the set b = a – 1 are: {(1, 0) (2, 1) (3, 2)}.

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When we speak of a relation between sets in math, we are talking about a set of ordered pairs in 2 dimensions or ordered triples in 3 dimensions.

The word ordered is the important one. It specifies that there is a set of first elements and a set of second elements. These sets are called the domain and the range of the relation.

 The domain of a relation is the set of all first members of the ordered pairs which make up the relation.The range is the set of all second members of the ordered pairs which make up the relation.

In the solution to example 4 above, the domain D = {1, 2, 3} and the range R = {0, 1, 2}.

As always in math, the words say it all. One's domain is the place where one lives or comes from. One's range is the places one goes to from one's domain, so it's easy to remember that we start with the domain set and go to the range set.

Example 5

E = { – 3, 1, 2,}

 (a) find the Cartesian product E × E (b) list the elements in the < relation (c) list the elements in the > relation (d) list the elements in the = relation (e) list the elements in the > (greater than or equal to) relation.

Solution

(a) E × E = {(3, 3) (3 ,1) (3, 2) (1, 3) (1, 1) (1, 2) (2, 3) (2, 1) (2, 2)}

(b) we want all the ordered pairs in which the 1st element is < the 2nd:

{(3 ,1) (3, 2) (1, 2)}

(c) we want all the ordered pairs in which the 1st element is > the 2nd:

{(1, 3) (2, 3) (2, 1))

(d) we want all the ordered pairs in which the two elements are equal:

{(3, 3) (1, 1) (2, 2)}

(e) now we want to combine the answers for (c) and (d):

{(2, 3) (2, 1) (3, 3) (1, 1) (2, 2)}

Another way to ask question (b) in this example is:
Find {(e1, e2) | e1 < e2 ; such that e1, e2 are elements in set E × E}

Example 6

List the Domain and Range for parts (c) and (d) in example 5.

Solution

(c) Domain = {1, 2}, Range = {3, 1}

(d) Domain = {3, 1, 2}, Range = {3, 1, 2}

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1) List the elements in set A × B where A = {1, 0, 1, 2,} and B = {5, 3, 1}

2) W = {1, 2, 3, 4}

a) List the ordered pairs in the Cartesian product W × W.
b) List the ordered pairs in your answer to (a) in which w2 = w1.
c) List the ordered pairs in your answer to (a) in which w2 > w1.
d) List the ordered pairs in your answer to (a) in which w2 = w1 + 2.

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1) A × B where A = {1, 0, 1, 2,} and B = {5, 3, 1}

A × B = {(1,5)(1,3)(1,1)(0,5)(0,3)(0,1)(1,5)(1,3)(1,1)(2,5)
(2,
3)(2,1)}

2) W = {1, 2, 3, 4}

a) List the ordered pairs in the Cartesian product W x W.

W × W = {(1,1)(1,2)(1,3)(1,4)(2,1)(2,2)(2,3)(2,4)(3,1)(3,2)(3,3)(3,4)(4,1)(4,2)(4,3)(4,4)}

b) List the ordered pairs in your answer to (a) in which w2 = w1.

{(1,1) (2, 2) (3, 3) (4, 4)}

c) List the ordered pairs in your answer to (a) in which w2 > w1.

{(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)}

d) List the ordered pairs in your answer to (a) in which w2 = w1 + 2.

{(1, 3) (2, 4)}

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