EXERCISE - A
a) An ordered pairs cannot have more than two components.
b) A set of ordered pairs should contain only two elements.
c) The cartesian products of any two sets is always a set of ordered pairs.
d) The cartesian product of sets is commutative.
e) Zero cannot be the second component of an ordered pair.
2) Find x and y in the following ordered pairs:
a) (2x, y)= (6,2).
b) (x, - y)= (2,-4).
c) (x + 1, y+ 2)= (4,5).
d) (x -2, y/2)= (1,2).
3) If A={1,2,3}, B={4,5} and C={1,2,3,4,5} find
a) A x B.
b) C x B.
c) B x B.
Hence show that (C x B) - (A x B)= (B x B).
4) x ∈ {2, 4,6,9} and y∈ {4, 6,18,27,54}. Form all ordered pairs (x,y) such that x is a factor of y and x < y.
5) Taking A={1,3,5,7} and B={2,4,6} Find
a) A x A
b) B x A
6) The table shows the coordinanate of Ax B as
(1,4) (2,4) (3,4)
(1,3) (2,3) (3,3)
(1,2) (2,2) (3,2)
(1,1) (2,1) (3,1)
Where A={1,2,3}, B={1,2,3,4}. State the number of ordered pairs which satisfy the relation.
a) is less than.
b) is greater than.
c) is equal to.
d) is one less than .
e) is two less than.
7) If P= {m,n} and Q={n,m}, then P x Q ={(m,n),(n,m)}. Is it true or false?
8) If {(a,x),(a,y),(b,x),(b,y)} is a product set or not?
9) If n(A)= x and n(B)= y and A∩ B= Φ, then n(A x B)= xy. Is it true or false.
10) If A and B are non empty sets, then A x B is a non empty set of ordered pairs (x,y) such that x ∈ B and y∈ A. Is it true or false?
11) Some elements of A x B are (a,x),(c,y),(d,z). If A={a,b,c,d}, find the remaining elements of A x B. Such that n(A x B) is least.
12) The ordered pairs (1,1),(2,2),(3,3) are among the elements in the set A x B . if A and B have 3 elements each , how many elements in all does the set A x B have. Also find the remaining elements.
13) Express each of the following as the set of the ordered pairs.
a) {(x,y),: 3x + 2y= 15; x, y ∈ N}.
b) {(x,y): 2x + y = 8; x,y ∈ W}.
c) {(x,y); x²+ y²= 4; x,y ∈ Z}.
14) Ordered pairs (5, m) and (n , -1) ∈ {(x,y): y= 2x -3}. Find the values of m and n.
15) If A={1,2,3,4}, B={5,7,9}, find (Ax B) ∩(A ∩B).
16) If A={1?2}, B={2,3}, C={3,4}, find
a) (A x B) U (A x C).
b) (A x B) ∩ (A x C).
RELATION: SYMMETRIC, REFLEXIVE AND EQUIVALENCE
RELATION (Intuitive Concept): The concept of relation is an association between two objects (people, numbers , ideas etc).
For Example :
1. Rama was son of Dashratha
2. Bangalore is the capital of Karnataka.
3. 15 is greater than 13.
4. {1,2} is a subset of {1, 2, 3}.
5. The line AB is parallel to the line CD.
Definition : (Mathematical concept). A relation is a set of ordered pairs. Any set of ordered pairs i.e., the cartesian product of any two sets is, therefore , a relation. The set of first components of the ordered pairs is called the domain and the set of second component is called the range.
it can also be defined as
A relation from set A to a set B is a subset of A x B. Thus if R is a relation from A to B, then R ⊂ A ⨯ B.
REPRESENTATION OF RELATION
A relation can be represented by several ways:
1. Roaster Form: By describing a set of ordered pairs.
In this form we record a set of ordered pairs, which satisfy a given relation.
2. Set Builder Form: By description of a rule
In this form we use a rule to describe a set of ordered pairs, in the form {(x,y) ∈ A x B : x .....y}, the blank is to be replaced by the rule.
3. By Tables as under:
Country : Sri Lanka England Japan India
Capital : Colombo London Tokyo New Delhi
4. By Arrow Diagrams
In this form , the relation is indicated by arrows drawn from the first component to the second.
5. By Graphs. The relation "is the capital of" can be represented by the graph, as in figure.
PROPERTIES OF RELATIONS
First Property - REFLEXIVE
A relation on a set A is called reflexive , if every element involved in in relation with itself, i.e., xRx, for all x ∈ A. Thus it contains ordered pairs (x,x) whose first component is the same as the second.
Second Property - Symmetric.
A relation on a set A is called symmetric if xRy and yRx are such that xRy => yRx i.e., (x,y) ∈ R => (y,x) ∈ R. Thus if x is related to y, then y is related to x.
Third Property= Transitive.
A relation R on a set A is called transitive if xRy and yRz => xRz.
Equivalence
A relation which is reflexive, symmetric and transitive is called an equivalence relation. Thus a relation R on a set A is called an equivalence relation, if it satisfies the following conditions :
i) xRx for all x ∈ A
ii) xRy => yRx ; x, y ∈ A
iii) xRy and yRz => xRz => xRz; x, y, z ∈ A.
EXERCISE - B
1) Draw the diagram of the relation "is cube of" on the set {1,2,3, 4, 8, 27, 64, 512}.
2) Let A={1, 2, 3,4} , B={ x, y, z}. Which of the following are relation from A to B ?
a) {1,y),(1 0,z),(3, x),(4, y)}. Y
b) {1,x),(2, y),(3, z),(1,4)}. No, because in the ordered pair (1,4), 4∈ A and not to B
c) {(1,x),(y, 4),( x, 3)}. No, because in (y,4), y ∉ A
d) {(y,1),(z, 1),( z,3 0),(y, 4)}. No, because here the first component of all ordered pairs are in the set B.
e) {(1,x),(x,1),(2,y),(y,2)}. N
3) Determine the domen and range of the following relations :
a) {(x, y): x is a multiple of 2 and y is a multiple of 3}.
b) {x, x²): x is a prime number < 11}
4) Show that the relation " ≡ congruence of the set of all triangles in Euclidean plane geometry in an equivalent relation.
5) Show that in the set of all real numbers, the relation "<" is transitive and anti-symmetric but not reflexive.
6) If E={(x,y): x ∈ N, y∈ N, (x - y) is divisible by 10}. Prove that R is an equivalence relation.
7) Prove that, " is parallel to" is an equivalence relation for a set of lines.
8)
MISCELLANEOUS
1) Write down the domain and range of the relation (x,y): x = 3y and x and y are natural numbers less than 10.
2) Given A={0, 1, 2, 3} and relation R define on set A, as
R={(x,y) ∈ A x A : x = y}; then
a) least element of R
b) List the domain of R
c) List the range of R.
3) A set of right cylinders is such that each has a height of 7cm. The radius of the bases of the cylinders varies, the maximum radius being 21cm. Let r represent the relations and v the volume of cylinders, where r, b ∈ R.
a) Define the relation between r and v as a set of order pairs.
b) State the domain of the relation.
c) State the range of the relation.
d) Is the relation a function ?
