ORDERED PAIR:
Elementary MathematicsORDERED PAIR:
An ordered pair (a, b) consists of two elements “a” and “b” in which “a” is the first element and “b” is the second element. The ordered pairs (a, b) and (c, d) are equal if, and only if, a= c and b = d. Note that (a, b) and (b, a) are not equal unless a = b.
EXERCISE:
Find x and y given (2x, x + y) = (6, 2)
SOLUTION: Two ordered pairs are equal if and only if the corresponding components are equal. Hence, we obtain the equations: 2x = 6 ………………(1) and x + y = 2 ……………..(2) Solving equation (1) we get x = 3 and when substituted in (2) we get y = -1.
ORDERED n-TUPLE:
The ordered n-tuple, (a_{1}, a_{2}, …, a_{n}) consists of elements a_{1}, a_{2}, ..a_{n} together with the ordering: first a_{1}, second a_{2}, and so forth up to a_{n}. In particular, an ordered 2- tuple is called an ordered pair, and an ordered 3-tuple is called an ordered triple.
Two ordered n-tuples (a_{1}, a_{2}, …, a_{n}) and (b_{1}, b_{2}, …, b_{n}) are equal if and only if each corresponding pair of their elements is equal, i.e., a_{i} = b_{j}, for all i = 1, 2… n.
CARTESIAN PRODUCT OF TWO SETS:
Let A and B be sets. The Cartesian product of A and B, denoted A × B (read “A cross B”) is the set of all ordered pairs (a, b), where a is in A and b is in B.
Symbolically: | ||
A ×B = {(a, b)| a ∈ A and b ∈ B} | ||
NOTE | ||
If set A has m elements and set B has n elements then | A ×B has m × n | |
elements. | ||
EXAMPLE: | ||
Let A = {1, 2}, B = {a, b, c} then | ||
A ×B = {(1,a), (1,b), (1,c), (2,a), (2, b), (2, c)} | ||
B ×A = {(a,1), (a,2), (b, 1), (b, 2), (c, 1), (c, 2)} | ||
A ×A = {(1, 1), (1,2), (2, 1), (2, 2)} |
B ×B = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b),(c, c)}
REMARK:
1. A × B≠B × A for non-empty and unequal sets A and B.
2. A ×φ = φ× A = φ
3. | A × B|= |A| × |B|
CARTESIAN PRODUCT OF MORE THAN TWO SETS:
The Cartesian product of sets A, A, …, A, denoted A× A× … ×A_{n}_{, }is the set of
12n12
all ordered n-tuples (a, a, …, a) where a_{1 }∈A, a∈A,…, a∈A n.
Symbolically: A× A× … ×A ={(a, a, …, a) |ai ∈Ai, for i=1, 2, …, n}
12n122n
12 n12n
BINARY RELATION:
Let A and B be sets. A (binary) relation R from A to B is a subset of A × B. When (a, b) ∈R, we say a is related to b by R, written a R b. Otherwise if (a, b) ∉R, we write a R b.
EXAMPLE:
Let A = {1, 2}, B = {1, 2, 3} Then A × B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Let
R_{1}={(1,1), (1, 3), (2, 2)} R_{2}={(1, 2), (2, 1), (2, 2), (2, 3)} R_{3}={(1, 1)} R_{4}= A × B R_{5}= ∅
All being subsets of A × B are relations from A to B.
DOMAIN OF A RELATION:
The domain of a relation R from A to B is the set of all first elements of the ordered pairs which belong to R denoted Dom(R). Symbolically: Dom (R) = {a ∈A| (a,b) ∈R}
RANGE OF A RELATION:
The range of A relation R from A to B is the set of all second elements of the ordered pairs which belong to R denoted Ran(R). Symbolically: Ran(R) = {b ∈B|(a,b) ∈ R}
EXERCISE:
Let A = {1, 2}, B = {1, 2, 3}, Define a binary relation R from A to B as follows: R = {(a, b) ∈A × B | a < b} Then
a.Find the ordered pairs in R.
b.Find the Domain and Range of R.
c.Is 1R3, 2R2?
SOLUTION:
Given A = {1, 2}, B = {1, 2, 3}, A × B = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3)}
a.R = {(a, b) ∈A × B | a < b} R = {(1,2), (1,3), (2,3)}
b.Dom(R) = {1,2} and Ran(R) = {2, 3}
a.Since (1,3) ∈R so 1R3 But (2, 2) ∉R so 2 is not related with3.
EXAMPLE:
Let A = {eggs, milk, corn} and B = {cows, goats, hens}
Define a relation R from A to B by (a, b) ∈R iff a is produced by b. Then R = {(eggs, hens), (milk, cows), (milk, goats)} Thus, with respect to this relation eggs R hens , milk R cows, etc.
EXERCISE :
Find all binary relations from {0,1} to {1}
SOLUTION:
Let A = {0,1} & B = {1}
Then A × B = {(0,1), (1,1)} All binary relations from A to B are in fact all subsets of A ×B, which are:
R_{1}= ∅ R_{2}={(0,1)} R_{3}={(1,1)} R_{4}={(0,1), (1,1)} = A × B
REMARK:
If |A| = m and |B| = n Then as we know that the number of elements in A × B are m × n. Now as we
m × n
know that the total number of and the total number of relations from A to B are2.
RELATION ON A SET:
A relation on the set A is a relation from A to A. In other words, a relation on a set A is a subset of A × A.
EXAMPLE: :
Let A = {1, 2, 3, 4} Define a relation R on A as (a,b) ∈ R iff a divides b {symbolically written as a | b} Then R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}
REMARK: For any set A
- A × A is known as the universal relation.
2. ∅ is known as the empty relation.
EXERCISE:
Define a binary relation E on the set of the integers Z, as follows: for all m,n ∈Z, m E n ⇔ m – n is even.
a.Is 0E0? Is 5E2? Is (6,6) ∈E? Is (-1,7) ∈E?
b.Prove that for any even integer n, nE0.
SOLUTION
E = {(m,n) ∈Z ×Z | m – n is even}
a. (i) (0,0) ∈ Z ×Z and 0-0 = 0 is even Therefore 0E0.
(ii) (5,2) ∈ Z ×Z but 5-2 = 3 is not even so 5 E 2
(iii) (6,6) ∈ E since 6-6 = 0 is an even integer.
(iv)(-1,7) ∈E since (-1) – 7 = -8 is an even integer.
a.For any even integer, n, we have n – 0 = n, an even integer so (n, 0) ∈E or equivalently n E 0
COORDINATE DIAGRAM (GRAPH) OF A RELATION:
Let A = {1, 2, 3} and B = {x, y} Let R be a relation from A to B defined as R = {(1, y), (2, x), (2, y), (3, x)} The relation may be represented in a coordinate diagram as follows:
A
EXAMPLE:
Draw the graph of the binary relation C from R to R defined as follows:
for all (x, y) ∈R × R, (x, y) ∈C ⇔ x^{2} + y^{2} = 1
SOLUTION
All ordered pairs (x, y) in relation C satisfies the equation x^{2}+y^{2}=1, which when solved for y gives
Clearly y is real, whenever –1 ≤ x ≤ 1
Similarly x is real, whenever –1 ≤ y ≤ 1
Hence the graph is limited in the range –1 ≤ x ≤ 1 and –1 ≤ y ≤ 1
The graph of relation is y
(0,1)
(1,0)
(-1,0)
(0,-1)
ARROW DIAGRAM OF A RELATION:
Let A = {1, 2, 3}, B = {x, y} and R = {1,y), (2,x), (2,y), (3,x)}
be a relation from A to B.
The arrow diagram of R is:
DIRECTED GRAPH OF A RELATION:
Let A = {0, 1, 2, 3}
and R = {(0,0), (1,3), (2,1), (2,2), (3,0), (3,1)}
be a binary relation on A.
DIRECTED GRAPH
MATRIX REPRESENTATION OF A RELATION
Let A = {a_{1}, a_{2}, …, a_{n}} and B = {b_{1}, b_{2}, …, b_{m}}. Let R be a relation from A to
B. Define the n × m order matrix M by
EXAMPLE:
Let A = {1, 2, 3} and B = {x, y} Let R be a relation from A to B defined as R ={(1,y), (2,x), (2,y), (3,x)}
EXAMPLE:
For the relation matrix.
1. List the set of ordered pairs represented by M.
2. Draw the directed graph of the relation.
SOLUTION:
The relation corresponding to the given Matrix is
• R = {(1,1), (1,3), (2,1), (3,1), (3,2), (3,3)} And its Directed graph is given below
EXERCISE:
Let A = {2, 4} and B = {6, 8, 10} and define relations R and S
from A to B as follows: for all (x,y) ∈A × B, x R y ⇔ x | y for all (x,y) ∈A × B, x S y ⇔ y – 4 = x
State explicitly which ordered pairs are in A × B, R, S, R∪S and R∩S.
SOLUTION
A × B = {(2,6), (2,8), (2,10), (4,6), (4,8), (4,10)} R = {(2,6), (2,8), (2,10), (4,8)} S = {(2,6), (4,8)} R ∪ S = {(2,6), (2,8), (2,10), (4,8)}= R R ∩ S = {(2,6), (4,8)}= S
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