## Review of Sets and Relations

Recall that a set is a well-defined collection of objects.

Examples:

a collection of flowers

{ Rose, Lily, Dahlia, Foxglove, Pentunia}

the set of natural numbers

N = { 1,2,3, ..............}

the set of integers

Z = { ........ -3,-2,-1,0,1,2,3, ............} etc.

The cartesian product of two sets A, B is A x B (read as A cross B), and is defined as

A x B= { (a,b) | a∈A, ∈B}

that is, A x B is a set of ordered pairs where the first coordinate belongs to the set A and the second
coordinate belongs to set B.

Consider this example.

Let A = {1,2,3} B = {-1,-2,3}

A x B = { (1,-1), (1,-2), (1,3), (2,-1), (2,-2), (2,3), (3,-1), (3,-2), (3,3) }.

If

n(A) = number of element in A = p

n(B) = number of elements in B = q

then n(A x B) = number of elements in A x B = p x q.

In the above case

n (A) = 3

n (B) = 3

n(A x B) = 3 x 3 = 9.

Consider these relations in mathematics.

1. a is equal to b or a=b

2. a is greater than b or a>b

3. line p is perpendicular to line q or p⊥ q

4. line x is parallel to y or x||y.

The symbols =, >,⊥ , || all denote specific relations. In general, we use R to denote a relation between two elements.
If x and y are in a relation, we denote it by x∈y or (x,y) ∈ R. If x and y are not in a relation, we denote this by x y or (x,y)
∉R. Sets have elements that are ordered pairs. In other words, a relation is a set of ordered pairs.

The property that connects the coordinates of an ordered pair in a relation is the formula of a relation.

Observe the following relations. State the formula, if any, for each of the following relations.

1. R1={(1,1), (2,2), (3,3)}

2. R2={(1,2), (1,3), (2,3)}

3. R3={(3,2), (3,1), (2,1)}

1. The coordinates of every ordered pair in R1 are equal to each other.

The formula for R1 is "equal to" or (a,b) ∈R1 ⇒a = b.

2. The coordinates of every ordered pair in R2 are less than.

The formula for R2 is "less than" or (a,b) ∈R2 ⇒a < b.

3. The coordinates of every ordered pair in R3 are greater than.

The formula for R3 is "greater than" or (a,b) ∈R3 ⇒a > b.

The set of the first coordinates of all ordered pairs of a relation is called the Domain of R. The set of the second coordinates of all ordered pairs of R is called the Range of R.

Consider R = {(1,2), (2,3), (3,4), (4,5)}

Domain of R = Dom R={1,2,3,4}

Range of R = {2,3,4,5}.

If R is a relation from set A into another set B, then by interchanging the first and second coordinates of R we get a new relation from B into A. This is called the Inverse Relation of R and is denoted by R-1.

So (x,y) ∈R if and only if (y,x) ∈R-1

if R = {(x,y) | x ∈A, y ∈B}

then R-1 = {(y,x) | y ∈B, x ∈A}

Domain of R-1 = range of R = B.

Range of R-1 = domain of R = A.

If R is the relation "is less than" then R-1 is the relation "is greater than".

Let R = {(a,b), (c,d), (e,f), (l,m)}

R-1 = {(b,a), (d,c), (f,e), (m,l)}

If R = ((a,a), (b,b), (c,c), (d,d)}

R-1 = {(a,a), (b,b), (c,c), (d,d)}

⇒ R = R-1.

#### Try these questions

1. If A = {12,13} B = {a,b}. Find A x B and show that n(A) * n(B) = n(A x B)

A x B = {(12,a), (12,b), (13,a), (13,b)}

n(A) = 2

n(B) = 2

n(A) ∗ n(B) = 2 ∗2 = 4

n (A x B) = 4

∴ n(A) ∗ n(B) = n(A x B)

2. If A = {2,4,6} B = {1,3,9}, R is defined as (x,y) ∉R 'x is greater than y'. Find R and (R-1)

R = {(2,1) (4,1)(4,3)(6,1)(6,3)}

R-1 = {(1,2) (1,4)(3,4)(1,6)(3,6)}

3. Write the domains and ranges of R and R-1 for the following relations.

R1 = {(1,2), (2,3), (3,4), (4,5)}

R2 = {(1,1), (2,4), (3,9), (4,16), (5,25)}

R3 = {(2,7), (3,8), (4,9), (5,10)}

1.    Domain of R1= {1,2,3,4}

Range of R1 = {2,3,4,5}

R1-1 = {(2,1), (3,2), (4,3), (5,4)}

Domain of R1-1 = {2,3,4,5}

Range of R1-1 = {1,2,3,4}

2.     Domain of R2= {1,2,3,4,5}

Range of R2= {1,4,9,16,25}

R2-1 = {(1,1), (4,2), (9,3), (16,4), (25,5)}

Domain of R2-1 = {1,4,9,16,25}

Range of R2-1 = {1,2,3,4,5}

3.     Domain of R3= {2,3,4,5}

Range of R3= {7,8,9,10}

R3-1 = {(7,2), (8,3), (9,4), (10,5)}

Domain of R3-1 = {7,8,9,10}

Range of R3-1 = {2,3,4,5}