EXERCISE-A
1) Prove the following:
a) the relation '>' is not symmetric.
b) The relation '≤' is transitive.
c) The relation 'is similar to' on the set T of all Triangles in a plane is transitive.
d) The relation 'is a square of' on the set of natural numbers is not Reflexive.
2) give an example of a relation which is:
a) only symmetric.
b) only Reflexive.
c) only transitive.
d) symmetrical and Reflexive and not transitive.
e) symmetric and transitive but not reflexive.
3) Decide in each of the following cases whether the relation is a) symmetric b) transitive c) reflexive. Justify by giving examples.
A) 'is greater than' on W.
B) 'is perpendicular to' on a set of lines in a plane.
C) 'is divisible by' on the set of real numbers.
D) 'is less than' on N.
E) ' is parallel to' on a set of lines.
F) ' is a multiple of'
4) Three relation R₁, R₂ and R₃are defined on a set A={a,b,c} as follows:
R₁= {(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}.
R₂={(a,a)}.
R₃={(b,c)}.
R₄={(a,b),(b,c),(c,a)}.
find whether or not each other relation R₁, R₂, R₃, R₄ on A is a)reflexive b) symmetric c) transitive.
5) if A={1,2,3,4} define relations on A which have properties of being
a) reflexive, transitive but not symmetric.
b) symmetric but neither reflexive nor transitive.
c) reflexive, symmetric and transitive.
6) Let R be a relation defined on the set of natural numbers N as R={(x,y): x,y ∈ N, 2x+ y= 41}.
Find the domain and range of R. Also, verify whether R is
a) Reflexive
b) Symmetric
c) Transitive.
7) Is it true that relation which is symmetric and transitive is also reflexive ? give reasons.
8) Test whether the following relations R₁, R₂, and R₃ are
A) Reflexive B) Symmetric C) Transitive
i) R₁ on Q₀ defined defined by (a,b)∈R₁ <=> a= 1/b.
ii) R₂ on Z defined by (a,b)∈ R₂ <=> |a- b| ≤ 5.
iii) R₃ on R defined by (a,b) ∈ R₃ <=> a² - 4ab + 3b²= 0.
9) An integer m is said to be related to another n if m is a multiple of n. check if the relation is symmetric, reflexive and Transitive.
10) Given the relation R={(1,2),(2,3)} on the set A={1,2,3}, add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and Reflexive.
11) let A= {1,2,3} and R={(1,2),(1,1),(2,3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.
12) give an example of a relation which is
A) reflexive and symmetric but not transitive,
B) reflexive and transitive but not symmetric.
C) symmetric and transitive but not reflexive.
D) symmetric but neither Reflexive nor Transitive.
E) transitive but neither Reflexive nor symmetric.
13) Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric.
14) the following relation are defined on the set of real numbers.
a) aRb if a- b > 0
b) aRb iff 1+ ab > 0
c) aRb if |a| ≤ b.
Find weather these relations are Reflexive, symmetric or Transitive.
15) Let A={1,2,3} and
R₁={(1,1),(1,3), (3,1),(2,2),(2,1),(3,3)};
R₂={(2,2),(3,1),(1,3)},
R₃={(1,3),(3,3)}. Find whether or not each of the relations R₁, R₂ R₃ on A is
A) reflexive
B) symmetric
C) transitive.
16) Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive.
A) R={(x,y): x and y work at the same place}.
B) R={(x,y): x and y live in the same locality}.
C) R={(x,y): x is wife of y}.
D) R={(x,y): x is father of y}
17) Check whether the relation R defined on the set A={1,2,3,4,5,6} as R={(a,b): b= a+1} is Reflexive, symmetric or Transitive.
18) Check whether the relation R on R defined by R={(a,b): a ≤ b³} is Reflexive, symmetric or transitive.
EXERCISE -2
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EQUIVALENCE RELATION
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1) the relation 'is sister of' on the set of the members of the family is not an equivalence relation.
2) Write the word 'yes' if the following relations are equivalence relation on the given set. write the word 'no' if it is not.
a) 'is equals to' on N. Yes
b) 'is a descendant' on a non- empty set of people. No
c) ' lives within 10 kilometres of' on a set or people who live in Delhi. No
d) " is the brother of" on the set of members of a family. No
e) 'is stronger than' on the set of children of class IX in a particular school. No
f) 'is divisible by' on the set of whole number(W). No
3) let M be the set of male members of a family and R means 'is brother of', prove that the relation R, over the set M, is an equivalence relation.
4) Let T be set of all triangles in a plane and the relation R means 'is similar to'. prove that R is an equivalence relation.
5) Show that for the set of all points in a plane, the relation 'at the same distance from the origin' is an equivalence relation.
6) Let I be the set of Integers and R be a relation on I x I defined as: R:{(a,b)∈ I x I : (a - b) is divisible by 7}.
Show that R is an equivalence relation.
7) a) Is the relation 'divides' on the set of positive integers an equivalence relation ? Prove it.
No, the relation is not symmetric.
b) Is the relation 'reciprocal of' on the set of non-zero real numbers an equivalence relation ? Prove it.
No, the relation is only symmetric
8) Prove that the relation ">" on the set of real numbers is not an equivalence relation.
9) show that the relation ' is subset of ' with respect to sets is not an equivalence relation.
10) Show that the relation '≤' on the set of Integers is not an equivalence relation.
11) Show that " is the father of" is not an equivalence relation.
12) In the set of all triangles in a plane, show that the relation " is congruent to" is an equivalence relation.
13) Show that the relation "≥" on the set of real numbers is not equivalence relation.
14) prove that "is parallel to" for a set of straight lines in a plane is an equivalence relation.
15) State which of these is an equivalence relation.
A) 'is greater than' on W.
B) 'is perpendicular to' on a set of lines in a plane.
C) 'is divisible by' on the set of real numbers.
16) A={real numbers}. On A, a relation k is defined by; for all a, b belongs to A, aRb holds if and only if the difference between a and b is less than 2. Is R an equivalence relation. Justify your answer.
17) state which of these is an equivalence relation.
a) 'is less than' on N.
b) ' is parallel to' on a set of lines.
c) ' is a multiple of'
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18) Prove that the relation R on Z defined by (a,b)∈ R <=> a - b is divisible by 5 is an equivalence relation on Z. 1
19) let n be a fixed positive integer. Define a relation R on Z as follows: (a,b)∈ R <=> a - b is divisible by n. Show that R is an equivalence relation on Z.
20) let S be a relation on the R of all real numbers defined by
S= {(a,b)∈ R x R: a²+ b²=1} prove that S is not an equivalence relation on R.
21) Let O be the origin. We define a relation between two points P and Q in a plane if OP= OQ. Show that the relation, so defined is an equivalence relation.
22) m is said to be related to n if m and n are integers and m - n is divisible by 13. does this define an equivalence relation ?
23) Let Z be the set of Integers. show that the relation R={(a,b): a,b ∈ Z and a + b is even} is an equivalence relation on Z.
24) Let Z be the set of all integers and Z₀ be the set of all non zero integers. Let a relation R on Z x Z₀ be defined as a follows:
(a,b)R(c,d) <=> ad = bc for all (a,b), (c,d) ∈ Z x Z₀, Prove that R is an equivalence relation on Z x Z₀
25) If R and S are relation on a set A, then Prove that
a) R and S are symmetric => R ∩ S and R U S are symmetric.
b) R is reflexive and S is any relation => R U S is reflexive.
26) R and S are transitive relation on a set A, then Prove that R U S may not be a transitive relation on A.
27) Show that the relation R on the set Z of integers, given by R={(a,b): 2 divides a- b}, is an equivalence relation.
28) Let R be the relation defined on the set both A={1,2,3,4,5,6,7} by R={a,b} : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7} are related to each other and all the elements of the subset {2,4,6}.
29) Show that the relation R on the set A={ x∈ Z ; 0≤ x ≤ 12}, given by R{(a,b): a= b}, is an equivalence relation. Find the set of all elements related to 1.
30) Show that the relation R, on the set A of all polygon
R={(P₁, P₂): P₁ and P₂ have same number of sides},
is an equivalence relation. What is the set of all elements in A related to the right angled triangle T with sides 3, 4 and 5 ?
Set of all triangles
31) Let L be the set of all lines in XY- plane and R be the relation in L defined as R={(L₁,L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4.
{y=2x+c: c ∈ R}
32) Show that the relation R defined by R={(a,b): a-b is divisible by 3 ; a,b ∈ N} is an equivalence relation.
EXERCISE ---3
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VERY SHORT ANSWER QUESTIONS
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1) Write the domain of the relation R defined on the set Z of integers as follows: (a,b)∈ R <=> a²+b²=25.
{0,±3,±4,±5}
2) If R={(x,y): x²+ y² ≤ 4, x,y ∈ Z} is a relation on Z, write the domain of R.
{0,±1,±2}
3) write the Identity relation on set A={a,b,c}. {a,a),(b,b),(c,c)}
4) write the smallest reflexive relation on set A{1,2,3,4}. {(1,1),(2,2),(3,3),(4,4)}
5) If R={(x,y): x+ 2y= 8} is a relation on N by, then write the range of if R. {1,2,3}
6) If R is symmetric relation on a set A, then write a relation between R and R⁻¹. R =R⁻¹
7) let R={(x,y): |x² - y²|<1} be a relation on the set {1,2,3,4,5}. Write R as a set of ordered pairs.
{(1,1),(2,2),(3,3),(4,4),(5,5)}.
8) If A={2,3,4}, B {1,3,7} and R={(x,y): x∈A, y ∈B and x< y} is relation from A to B, then write R⁻¹. {(3,2),(7,2),(7,3),(7,4)}
9) let A={3,5,7}, B{2,6,10} and R be a relation from A to B defined by R={(x,y): x and y are relatively prime}. Then, write R and R⁻¹. R= {(3,2),(3,10),(5,2),(5,6),(7,2),(7,6),(7,10)}
R⁻¹={(2,3),(10,3),(2,5),(6,5),(2,7), (6,7),(10,7)
10) If A={3,5,7} and B{2,4,9} and R is a relation given by " is less than", write R as a set ordered pairs.
{(3,4),(3,9),(5,9),(7,9)}
11) A{1,2,3,4,5,6,7,8} and if R={(x,y): y is one half of x; x,y ∈A} is a relation on A, then write R as a set of ordered pairs.
{(2,1),(4,2),(6,3),(8,4)}
12) let A={2,3,4,5} and B{1,3,4}. If R is the relation from A to B given by a R b iff " a is a divisor of b". write R as a set of ordered pairs. {(2,4),(4,4),(3,3)}
EXERCISE --4
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1) " is greater than" for the set of real numbers is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
2) is the cube of" for the set of real numbers is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
3) 'is the sister of" for the set of real numbers is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
4) 'is similar to" for the set of real numbers is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
5) is perpendicular to" for the set of real numbers is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
6) Write down a relation which is
only Transitive.
A) is greater than
B) is perpendicular to
C) is a multiple of
D) is a friend of
7) Write down a relation which is
only symmetric.
A) is greater than
B) is perpendicular to
C) is a multiple of
D) is a friend of
8) Write down a relation which is
only Reflexive and Transitive.
A) is greater than
B) is perpendicular to
C) is a multiple of
D) is a friend of
9) Write down a relation which is
only symmetric and reflexive
A) is greater than
B) is perpendicular to
C) is a multiple of
D) is a friend of
10) If A is the set of the members of a family and R means "is brother of" then it is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
11) The relation R in the set {1,2,3} given by R={(1,2),(2,1) is
A) symmetrica relation
B) Transitive relation
C) Reflexive relation
D) equivalence relation D) none
12) The relation R in the set {1,2,3,4} given by R={(1,2),(2,2),(1,1), (4,4), (1,3), (3,3),(3,2)} is
A) R is reflexive and symmetric but not Transitive
B) R is reflexive and transitive but not symmetric
C) R is symmetric and Transitive but not reflexive
D) R is an equivalence relation
13) The relation R in the set A of real numbers defined as R= {(a,b): a≤ b} is
A) R is reflexive and symmetric but not Transitive
B) R is reflexive and transitive but not symmetric
C) R is symmetric and Transitive but not reflexive
D) R is an equivalence relation
14) The relation R in the set of real numbers defined as R={(a,b): a≤ b²} is
R is reflexive and symmetric but not Transitive
B) R is neither reflexive nor transitive nor symmetric
C) R is symmetric and Transitive but not reflexive
D) R is an equivalence relation
15) In the set of all triangles in a plane, the relation of similarity is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
16) Is the the relation "is the square of" for the set of natural numbers N is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
E) none
17) "is smaller than"
A) symmetric
B) transitive
C) reflexive
D) equivalence relation E) none
18) "is the father of"
A) symmetric
B) transitive
C) reflexive
D) equivalence relation E) none
19) "is parallel to" for set of straight lines.
A) symmetric
B) transitive
C) reflexive
D) equivalence relation E) none
20) "is a multiple of" for a set of positive integers
A) symmetric
B) transitive
C) reflexive
D) equivalence relation E) none
21) "is congruent to"
A) symmetric
B) transitive
C) reflexive
D) equivalence relation E) none
22) "is congruent to" in a set of triangles is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
23) If R is a relation in N x N defined by (a,b) R (c,d) if only of a+ d = b+c, then R is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
24) The relation R in the set A={1, 2, 3, 4,5} given by R={(a,b): | a - b | is even} is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
25) Let I be the set of all integers and R be the relation on I defined by a R b iff (a+b) is an even integer for all a, b belongs to I. Then it is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
26) Let I be the set of all integers and R be the relation on I defined by R={(x,y): x,y belongs to I, x- y is divisible by 11} is
A) symmetric
B) transitive
C) reflexive
D) equivalence relation
27) The relation R in the set {1,2,3} given by R={(1,2),(2,1) is
A) symmetrica and Transitive
B) symmetric and reflexive
C) symmetric, Reflexive and Transitive.
D) symmetric but neither reflexive nor Transitive.
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