RELATION__
Let A and B be two non-empty sets. Then, a relation R from A to B is a subset of (A x B).
Thus, R is relation from A to B
<=< R ⊂ (A x B).
If (a,b) ∈ R then we say that 'a is related to B' and we write, a R b.
If (a, b) not belongs to R then 'a is not related to b'.
** Let R be a relation from A to B. Then, R ⊂ (A x B).
DOMAIN:
The set of all first coordinates of elements of R is called the domain of R, written as Dom (R).
RANGE:
The set of all second coordinates of elements of R is called the range of R, denoted by range (R).
CO-DOMAIN:
The set B is called the coordinates of R.
Thus, Dom(R)= {a : (a,b) ⊂ R} and
Range (R)= {b : (a,b) ∈ R}
TYPES OF RELATION:
Let A be a non-empty set. Then, a relation R on A is said to be:
i) Reflexive: if, (a,a) ∈ R for all ∈ A, i e., a R a for all ∈ A.
ii) symmetric: if, (a, b) ∈ R
=> (b,a) ∈ R for all a,b ∈ A, i.e, a R b
=> b R a for all a, b ∈ A.
iii) Transitive: if (a,b) ∈ R and (b, c) ∈ R => (a,c) ∈ R for all a, b, c ∈ A, i.e., a R b, b R c
=> a R c for all a, b, c ∈ A
• EQUIVALENCE RELATION:
A relation which is Reflexive, symmetric and transitive, is called an equivalence relation.
EXERCISE-1
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1) Let A={3,4,5,6} and B={3,4,5}, find A x B, then find the set of all ordered pairs which satisfy:
A) is equals to. (3,3),(4,4),(5,5)
B) is less than. (3,4),(3,5),(4,5)
C) is greater than. (4,3),(5,3),(5,4), (6,3),(6,4),(6,5)
D) is two less than. (3,5)
E) is two more than. (5,3),(6,4)
F) is a factor of. (3,3),(4,4),(5,5)
2) If A={2,3,4,5,6,7,8,9}, find the set of elements of relations. L and M on the set A such that.
a) L means "is Square of". {(4,2)(9,3)}
b) M={(x,y): x+y=8}. {(2,6),(3,5),(4,4),(5,3),(6,2)
3) Let A{a,b,c,d} and B{x,y,z}. State which of the following are relations from A to B:
a) {(a,y),(a,z),(c,x),(d,y)}.
b) {(a,x),(a,y),(a,z),(b,x)}
c) {(x,b),(y,a),(y,c),(z,d)}. a,b
4) Given A={a,b,c,d} and B={x,y,z}. state which of the following are relations from B to A:
a) {(a,x),(b,y),(c,z),(d,x)}
b) {(x,a),(y,b),(z,c),(z,d)}
c) {(a,b),(b,z),(x,a),(z,b)}
d) {(x,a),(x,c),(y,a),(z,b),(z,d)}. b,d
5) Given relation R={(x,y): x> y+2}. State which of the given ordered pairs belong to this relation.
a) (3,0)
b) (5,4)
c) (-7,-10)
d) (7,9). a,c
6) Write the domain and range of the relation:
a) {(3,-3),(4,-4),(5,-5),....}.
D:{3,4,5.....}
R:{-3,-4,-5,.....}
b) {(-3,1),(-2,1), (-1,1), (1,1),(2,1), (3,1)}.
D:{-3,-2,-1,1,2,3}
R:{1}
c) (1,1),(2,8),(3,27),....}
D:{1,2,3,......}
R:{1,8,27.........}
d) {2, 1/2), (3,1/3), (4,1/4),(5,1/5).
D:{2,3,4,5}
R:{1/2,1/3,1/4,1/5}
7) Find the domain and the range of the relation:
a) {(x, 1/x): x ∈ N and 2< x ≤ 6}.
D:{3,4,5,6}
R:{1/3,1/4,1/5,1/6}
b) {(x,y): x,y ∈N and x+y < 6}.
D:{1,2,3,4}
R:{1,2,3,4}
c) {x, x-2): x ∈N and 2< x² ≤ 50}.
D:{2,3,4,5,6,7}
R:{0,1,2,3,4,5}
8) Given A={1,2,3,5,7,8,9,16,27,36}. find the relation R on A, when R means "is cube of".
D:{(1,1),(8,2),(27,3)
9) If A={3,5,7,8} & B={4,5,6,7,8,9}. Then find:
i) K={(a,b)∈ A x B : a-b=1}. {(5,4),(7,6),(8,7)}
ii) L={(a,b)∈ A xB : a=b-2}. {(3,5),(5,7),(7,9)}
iii) M={(a,b)∈ A xB : 2a+3=b} {(3,9)}
iv) N={(a,b)∈ A xB : a> b}. {(5,4),(7,4),(7,5),(7,6),(8,4),(8,5),(8,6),(8,7)
10) For each of the following relations, determine the domain and range:
a) {(x,y): y= x², -2≤x≤3, x∈ Z}.
D:{-2,-1,0,1,2,3}
R:{0,1,4,9}
b) {(x,y): x= y+ √(4-y²), -2≤y≤2, y∈ Z}
D:{-2,-1+√3,2,1+√3}
R:{-2,-1,0,1,2}
c) {(x,y): y= 3x²+2, x≤3, x∈ W}.
D:{0,1,2,3}
R:{2,5,14,29}
11) given A={4,5,6,7,8,9}. List all elements of:
I) {(x,y): ∈ A x A: x> y and x/y not ∈ to N}. (5,4),(6,4),(6,5),(7,4),(7,5),(7,6),(8,5),(8,6),(8,7),9,4),(9,5),(9,6),(9,7) and (9,8)
ii) {(x,y): ∈ A x A: x≠ y and y/x ∈ N}
(4,8)
12) Let A={4,5,6,7,8,9}; B={ a∈Z: -3≤ a ≤7} and a relation R in A x B such that:
R= {(x,y): y= x-5}. Find
I) the elements of R. (4,-1),(5,0)(6,1),(7,2),(8,3) and (9,4)
ii) domain and range of R. D:{4,5,6,7,8,9}
R:{-1,0,1,2,3,4}
iii) the range of variable x. Set B
iv) the domain of variable x. Set A
13) If A={1,3,5,7......} and
B={1,4,9,16,25 .....}, represent the relation from A to B {(1,1),(3,9),(5,25),(7,49),....} by means of an arrow diagram. Find the domain and range. D:{3,6,9}, R:{1,2,3}
14) Write the domain and range of the relation {(x,y): 3y= x}, where x and y are natural numbers less than 10}.
15) given A={x ∈N: 2≤ x≤ 16}. list the elements of the following relations:
I) {(x,y): ∈ A x A: x+ y≤8}. {(2,2),((2,3),(2,4),(2,5),(2,6),(3,2),(3,3),(3,4),(3,5),(4,2),(4,3),(4,4),(5,2),(5,3),(6,2)}
ii) {(x,y): ∈ A x A: x=√ y}. {(2,4),(3,9),(4,16)}
iii) {(x,y): ∈ A x A: 2x+ y>15}. {(2,12),(2,13),(2,14),(2,15),(2,16),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),....... So on..
iv) {(x,y): ∈ A x A: x≠ y and x/y ∈ N}
(4,2),(6,2),(8,2),(10,2),(12,2),(15,3),(8,4),(12,4),(16,4),(10,5),(14,5),(12,6),(14,7),(16,8)
16) Let A={0,1,2,3,4,5,}
B={-5,-4,-3,-2,-1,0,1,2,3,4} and
S= {(x,y): ∈ A x B: y= 2x-5}.
I) list the elements of S. (0,-5),(1,-3),(2,-1),(3,1),(4,3)
ii) list the domain of S. {0,1,2,3,4}
iii) list the range of S. {-5,-3,-1,1,3}
iv) What is the domain of the variable x ? Set A
17) Find the linear relation between the components of the ordered pairs of the following relations:
I) {(2,0),(3,3),(4,6),(5,9)}
ii) {(0,2),(1,3),(2,4,),(3,5,)}. y= x+2
iii) {(2,5),(3,7),(4,9),....}. y= 2x+1
18) Express each of the following relation as the set of ordered pairs:
I) = {(x,y): 2x+y= 8; x,y ∈ W}. {(0,8),(1,6),(2,4),(3,2),(4,0)}
ii) {(x,y): x²+y²= 25; x,y ∈ W}. {(0,5),(3,4),(4,3),(5,0)}
iii) {(x,y): x²+y²= 4; x,y ∈ Z}. {(-2,0),(2,0),(0,-2),(0,2)}
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** Let R be a relation from A to B, Then, R is subset of (A x B).
1) The set of all first co-ordinates of elements of R is called the domain of R, written as dom(R).
2) The set of second coordinates of elements of R is called the range of R, denoted by range (R).
3) The set B is called the co-domain of R.
Thus dom(R)={a:(a,b) subset of R}
And range (R)={b:(a,b∈ R}.
** ARROW DIAGRAM:
Let R be a relation from A to B.
We puts dots to represent the elements of A and B.
For each (a,b) ∈ R, we draw an arrow from a to b.
++++++++()+++++++()++++++++
Exercise - 2
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1) Find x and y, if:
A) (x+3,5)=(6, 2x+y). 3, -1
B) (a/3 +1, b- 2/3)=(5/3,1/3). 2,1
C) (x+1,1)= 3, y-2). 2,3
2) If the ordered pairs (x,1) and (5,y) belongs to the set {(a,b): b=2a -3}, find the value of x, y. 2,7
3) If a ∈{2,4, 6,9} and b ∈ {4,6,18,27} then form the set of all ordered pairs (a,b) such that a divides b and a< b. {(2,4),(2,6),(2,18),(6,18), (9,18),(9,27)}
4) Let A={1,2,3} & B{3,4}. Find AxB and B xA. Show it graphically.
5) Let A={1,2} & B{1,3}. Find AxB and B xA. Show it graphically.
6) If A={1,2,3} & B{2,4}. What are
A) AxB
B) B xA
C) Ax A
D) Bx B
E) (AxB)∩(BxA)
7) If A and B are two sets having 3 elements in common if n(A)= 5, n(B)= 4, find
A) n(AxB). 20
B) n{(AxB)∩(BxA)}. 9
8) Let A and B be two sets such that n(A)= 3 and n(B)= 2.
If (x,1), (y,2), (z,1) are in AxB, Find A and B, where x, y, z are distinct elements. A={x,y,z}, B={1,2}
9) If A={1,2}, form the set of AxAxA.
10) If A={1,2,4} and B={1,2,3}, represent following sets graphically:
A) AxB
B) Bx A
C) AxA
D) B x B
11) Let A={1,2,3} and B={2,4,6}. Show that R={(1,2),(1,4),(3,2),(3,4) is a relation from A to B. Find
A) dom(R). {1,3}
B) co-domain (R). {2,4,6}
C) range (R). {2,4}
D) Depict the above relation by an arrow diagram.
12) Let A={1,2,3,4,5} & B={1,4,5}
Let R be a relation ' is less than' from A to B.
A) list the element of R. {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)}
B) domain. {1,2,3,4}
C) co-domain. {1,4,5}
D) range. {4,5}
E) Depict the above relation by an arrow diagram.
13) Let A={1,2,3,4,5,6} Define a relation R on A by R={(x,y): y=(x+1)}.
A) Depict R, using Arrow diagram.
B) Domain. {1,2,3,4,5}
C) co-domain. {1,2,3,4,5}
D) range. {2,3,4,5,6}
14) Let A be the set of first 10 natural numbers and let R be a relation on A defined by (x,y) ∈ R <=> x+ 2y= 10,
i.e R={(x,y):x∈ A, y ∈ A & x+ 2y= 10}. Express R and R⁻¹ as sets of ordered pairs. Determine also
A) domain of R and R⁻¹.
B) ranges of R and R⁻¹.
15) A relation R is defined from a set A={2,3,4,5,} to a set B={3,6,7,10} as follows: (x,y)∈ R <=> x divides y. Express R as a set of ordered pairs and determine the domain and range of R . Also find R⁻¹.
{(2,6),(2,10),(3,3),(3,6),(5,10)}
Dom:- {2,3,5}, Range:- {3,6,10}
R⁻¹={(6,2),(10,2),(3,3),(6,3),(10,5}
15) If R is the relation ' less than ' from A={1, 2, 3, 4, 5} to B={1,4,5}, write down:
A) the set ordered pairs corresponding to R. {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)}
B) Find the inverse R. {(4,1),(5,1), (4,2),(5,2),(4,3),(5,3),(5,4)}
16) A relation R is defined on the set Z of integers as follows:
(x,y) ∈ R <=> x²+ y²= 25.
Express R and R⁻¹as the set of ordered pairs and hence find their respective domains.
R={(0,5),(0,-5),(3,4),(-3,4),(3,-4), (-3,-4),(4,3),(-4,3),(4,-3),(-4,-3),(5,0), (-5,0)
R⁻¹={(5,0),(-5,0),(4,3),(4,-3),(-4,3), (-4,-3),(3,4),(3,-4),(-3,4),(-3,-4),(0,5), (0,-5)
Domain={0,3,-3,4,-4,5,-5}= domain of R⁻¹.
17) Let R be the relation on the set of N of natural numbers defined by R={(a,b): a+ 3b= 12, a ∈ N, b ∈ N}, Find
A) R. {(9,1),(6,2),(3,3)}
B) Domain. {9,6,3}
C) range. {1,2,3}
18) Let A={1,2,3,4,5,6}. Define a relation R on set A by R={(x,y): y= x+1}
A) Depict this relation using an arrow diagram.
B) write down the domain, co-domain and range of R.
Dom:-{1,2,3,4,5}
Range:- {2,3,4,5,6}
19) A relation R is defined from a set A={2,3,4,5} to a set B={3,6,7,10} as follows:
(x,y) ∈ R <=> is relatively prime to y.
Express R as a set of ordered pairs and determine its
A) domain.
B) range.
20) Let A be the set of first five natural numbers and let R be a relation on A defined as follows: (x,y)∈ R <=> x ≤ y
Express
A) R and R⁻¹ as set of ordered pairs. {(1,1),(1,2),(1,3),(1,4),(1,5), (2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5), (4,4),(4,5),(5,5)}
R⁻¹ {(1,1),(2,1),(3,1), (4,1),(5,1),(2,2),(3,2),(4,2),(5,2), (3,3),(4,3),(5,3),(4,4),(5,4)}
B) Domain of R⁻¹. {1,2,3,4,5}
C) the range of R. {1,2,3,4,5}
21) Find the inverse relation R⁻¹in each of the following cases:
A) R={(1,2),(1,3),(2,3),(3,2),(5,6). {(2,1),(3,1),(3,2),(2,3),(6,5)}
B) R={(x,y): x,y ∈ N, x+ 2y= 8}. {(3,2),(2,4),(1,6)}
C) R is a relation from {11, 12, 13} to {8, 10, 12} defined by y= x -3. {(8,11),(10,13)}
22) let A={1, 2, 3,... 14}. define a relation on a set A by
R={(x,y): 3x-y= 0, where x,y ∈ A,
Depict this relationship using arrow diagram. write down its
A) domain. {1,2,3,4}
B) co-domain. A
C) range. {6,7,8}
23) Define relation R on the set N of natural numbers by R==(x,y): y=x+5, x is a natural number less than 4, x,y ∈ N}. Depict this relationship using
A) roster form. {(1,6),(2,7),(3,8)}
B) an arrow diagram.
C) domain. {1,2,3}
D) range. {6,7,8}
24) A={1,2,3,5} and B={4,6,9}. define a relation R from A to B by R={(x,y): the difference between x and y is odd, x∈ A, y∈ B}. write R in roster form. {(1,4),(1,6),(2,9),(3,4),(3,6), (5,4),(5,6).
MULTIPLE CHOICE QUESTIONS
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1) If A={1,2,4}, B={2,4,5}, C{2,5}, then (AxB)x (B - C) is
A) {(1,2),(1,5),(2,5)} B) {(1,4)}.
C) (1,4) D) none
2) If R is a relation on the set A={1, 2, 3,4,5,6,7,8,9} given by x R y <=>y = 3x, then
A) {(3,1),(6,2),(8,2),(9,3).
B) {(3,1),(6,2),(9,3)}
C) {(3,1),(2,6),(3,9)} D) none.
3) Let A={1,2,3},.B={1,3,5}. If a relation R from A to B is given by R ={(1,3),(2,5), (3,3)}. Then R⁻¹ is
A) {(3,3),(3,1),(5,2)}.
B) {(1,3),(2,5),(3,3)}
C) {(1,3),(5,2)} D) none
4) If A={1,2,3}, B={1,4,6,9} and R is a relation from A to B to defined by ' x is greater than y'. the range of R is
A) {1,4,6,9} B) {4,6,9} C) {1}. D) n
5) if R={(x,y): x, y ∈ Z, x²+ y²≤ 4} is a relation on Z, then domain of R is
A) {0,1,2} B) {0,-1,-2}
C) {-2,-1,0,1,2}. D) none
6) A relation R is defined from {2,3,4,5} to {3,6,7, 10} by: x R y <=> x is relatively Prime to y. Then domain of R is
A) {2,3,5} B) {3,5}
C) {2,3,4} D) {2,3,4,5}.
7) A relation from ¢ from C to R is Defined by x ¢ y <=> |x| = y. Which one is correct ?
A) (2+3i) ¢13 B) 3 ¢ (-3)
C) (1+ i)¢ 2 D) i ¢ 1.
8) Let R be a relation on N Defined by x+2y = 8. The domain of R is
A) {2,4,8} B) {2,4,6,8}
C) {2,4,6}. D) {1,2,3,4}
9) R is a relation from {11,12, 13} to {8, 10, 12} Defined by y= x - 3. then inverse of R is
A) {(8,11), (10,13)}.
B){(11,8),(13,10)}
C) {(10,13),(8,11),(12,10)} D) n
10) If the set A has p elements, B has q elements, then the number of elements in A x B is
A) p+q B)p+q+1 C) pq. D) p²
11) Let R be a relation from a set to B, then
A) R = A U B B) R= A ∩ C
C) R ⊂ A x B. D) R ⊂ B x A
12) If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relation from A to B is
A) 2ᵐⁿ. B) 2ᵐⁿ -1 C) 2mn D) mⁿ
13) If R is a relation on a finite set having n elements, then the number of relation on A is
A) 2ⁿ B) 2ⁿ^². C) n² D) nⁿ
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