Let N be the set of natural numbers and R be the relation on N × N defined by ( a , b ) R ( c , d ) iff a d = b c for all a , b , c , d ∈ N . Show that R is an equivalence relation.

Given that, N is the set of natural numbers and R be the relation on N × N defined by ( a , b ) R ( c , d ) iff a d = b c for all a , b , c , d ∈ N We know that, a relation R that is reflexive, symmetric and transitive is called an equivalence relation. Reflexive: For any ( a , b ) ∈ N × N a · b = b · a ∴ ( a , b ) R ( a , b ) Thus, R is reflexive Symmetric: For any ( a , b ) , ( c , d ) ∈ N × N ( a , b ) R ( c , d ) ⇒ a · d = b · c ⇒ c · b = d · a ⇒ ( c , d ) R ( a , b ) Thus, R is symmetric Transitive: For any ( a , b ) , ( c , d ) , ( e , f ) , ∈ N × N ( a , b ) R ( c , d ) and ( c , d ) R ( e , f ) ⇒ a · d = b · c and c · f = d · e ⇒ a · d · c · f = b · c · d · e ⇒ a · f = b · e ∴ ( a , b ) R ( e , f ) Thus, R is transitive Therefore, R is an equivalence Relation.

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.

Match each disease with its correct type of vaccine: (a) Tuberculosis (i) Harmless virus (b) Whooping cough (ii) Inactivated toxin (c) Diphtheria (iii) Killed bacteria (d) Polio (iv) Harmless bacteria

(a) (b) (c) (d) (iv) (iii) (ii) (i)

(a) (b) (c) (d) (ii) (i) (iii) (iv)

(a) (b) (c) (d) (iii) (ii) (iv) (i)

(a) (b) (c) (d) (i) (ii) (iv) (iii)

Show that the relation R on R is defined as R = { ( a , b ) : a ≤ b } , is reflexive and transitive but not symmetric.

Given, R = a , b : a ≤ b , a , b ∈ R For the relation R to be reflexive: If the relation R is reflexive, then a , a ∈ R i.e. a ≤ a , which is always true for all the values of a , a ∈ R So, the given relation R is reflexive. For the relation R to be symmetric: If a , b ∈ R then b , a ∈ R i.e. if a ≤ b then b ≤ a , which is not possible for all the values of a , b ∈ R . For example: if a = 2 and b = 5 Here, a < b but b > a So, the given relation R is not symmetric. For the relation R to be transitive: If a , b ∈ R and b , c ∈ R then a , c ∈ R i.e.,if a ≤ b and b ≤ c then a ≤ c , which is true for all the values of a , b , c ∈ R . For example: if a = 2 , b = 3 and c = 4 Here, a < b < c So, the given relation is transitive. Hence, the given relation R is reflexive and transitive but not symmetric.

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Which organization was established as a collective defense arrangement to counter the Soviet Union during the Cold War?

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Important Questions on Relations and Functions

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

The ratio of mass densities of nuclei of

^{40} C a

and

^{16} O

is close to:

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

For

x ϵ (0, \frac{3}{2}),

let

f (x) = \sqrt{x}, g (x) = t a n x

and

h (x) = \frac{1 - x^{2}}{1 + x^{2}}

. If

ϕ (x) = (h o f) o g) (x),

then

ϕ (\frac{π}{3})

is equal to:

MEDIUM

Mathematics>Algebra>Relations and Functions>Composition of Functions

2 M e V

neutron is emitted in a fission reactor. If it looses half of its kinetic energy in each collision with a moderator atom, how many collisions must it undergo to achieve thermal energy of

0.039 e V

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

Antivenom injection contains preformed antibodies while polio drops that are administered into the body contain

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

A relation

R

in a set

A

is called _____, if

(a_{1}, a_{2}) \in R

implies

(a_{2}, a_{1}) \in R

, for all

a_{1}, a_{2} \in A

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

Asthma may be attributed to:

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

f (x) = x + 1

, find

\frac{d}{d x} (fof) (x)

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

The main barrier that prevents the entry of micro-organisms into our body is

HARD

Mathematics>Algebra>Relations and Functions>Composition of Functions

Let

N

be the set of natural numbers and

R

be the relation on

N \times N

defined by

(a, b) R (c, d) iff a d = b c

for all

a, b, c, d \in N

. Show that

R

is an equivalence relation.

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

Check whether the relation

R

defined in the set

\{1, 2, 3, 4, 5, 6\}

R = \{(a, b) : b = a + 1\}

is reflexive, symmetric or transitive.

MEDIUM

Mathematics>Algebra>Relations and Functions>Composition of Functions

Match each disease with its correct type of vaccine:

(a)	Tuberculosis	(i)	Harmless virus
(b)	Whooping cough	(ii)	Inactivated toxin
(c)	Diphtheria	(iii)	Killed bacteria
(d)	Polio	(iv)	Harmless bacteria

MEDIUM

Mathematics>Algebra>Relations and Functions>Composition of Functions

Show that the relation

R

R

is defined as

R = {(a, b) : a \leq b}

, is reflexive and transitive but not symmetric.

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

MALT constitutes about _______ percent of the lymphoid tissue in human body.

HARD

Mathematics>Algebra>Relations and Functions>Composition of Functions

Explain the principle and working of a nuclear reactor with the help of a labelled diagram.

MEDIUM

Mathematics>Algebra>Relations and Functions>Composition of Functions

How do cytokine barriers provide innate immunity in humans?

HARD

Mathematics>Algebra>Relations and Functions>Composition of Functions

Comment on the role of vaccination and immunisation in keeping the human population healthy.

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

Name any two physiological barriers that provide innate immunity.

MEDIUM

Mathematics>Algebra>Relations and Functions>Composition of Functions

Let

R

be a relation defined on

ℕ

as a

R

b is

2 a + 3 b

is a multiple of

5, a, b \in ℕ

. Then

R

HARD

Mathematics>Algebra>Relations and Functions>Composition of Functions

Let

A = {x \in Z : 0 \leq x \leq 12}

. Show that

R = {(a, b) : a, b \in A, | a - b | is divisible by 4}

is an equivalence relation. Find the set of all elements related to

1

. Also write the equivalence class

[2]

EASY

Mathematics>Algebra>Relations and Functions>Composition of Functions

Consider the non-empty set consisting of children in a family and a relation

R

defined as

a R b

a

is brother of

b

. Then

R

Which organization was established as a collective defense arrangement to counter the Soviet Union during the Cold War?

Important Questions on Relations and Functions

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