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.

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.

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 .

We have, n a fixed positive integer and relation R on Z as ( a , b ) ∈ R ⇔ a − b is divisible by n Let a ∈ R , t hen a − a = 0 = 0 × n ⇒ a − a is divisible by n ⇒ ( a , a ) ∈ R for all a ∈ Z So , R is reflexive on Z Let ( a , b ) ∈ R Here , a − b is divisible by n ⇒ a − b = n p for some p ∈ Z ⇒ b − a = n ( − p ) ⇒ b − a is divisible by n [ p ∈ Z ⇒ − p ∈ Z ] ⇒ ( b , a ) ∈ R So , R is symmetric on Z Let ( a , b ) and ( b , c ) ∈ R Here , a − b is divisible by n and b − c is divisible by n . ⇒ a − b = n p for some p ∈ Z and b − c = n q for some q ∈ Z Adding both, we get a − b + b − c = n p + n q ⇒ a − c = n ( p + q ) ⇒ ( a , c ) ∈ R for all a , c ∈ Z So , R is transitive on Z Hence, R is an equivalence relation on Z .

MEDIUM

Earn 100

Which agreement was signed between India and China in 1993 to maintain peace along the Line of Actual Control?

50% studentsanswered this correctly

Important Questions on Nuclei

MEDIUM

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Four nuclei of an element undergo fusion to form a heavier nucleus, with release of energy. Which of the two the parent or the daughter nucleus would have higher binding energy per nucleon ?

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Out of

{[{CoF}_{6}]}^{- 3}

and

[Co {(en)}_{3}]^{3 +}

, which complex compound more stable.

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

The ratio of mass densities of nuclei of

^{40} C a

and

^{16} O

is close to:

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

HARD

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

HARD

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Explain the processes of nuclear fission and nuclear fusion by using the plot of binding energy per nucleon

(B E / A)

versus the mass number

A

MEDIUM

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Show that the relation

R

R

is defined as

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

, is reflexive and transitive but not symmetric.

HARD

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

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

MEDIUM

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Let

A = {1, 2, 3}

, and let

R_{1} = \{(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)\}, R_{2} = \{(2, 2), (3, 1), (1, 3)\}, R_{3} = \{(1, 3), (3, 3)\}

. Find whether or not each of the relations

R_{1}, R_{2}, R_{3}

A

A = \{1, 2, 3\}

is reflexive, symmetric and transitive.

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

The following relation is defined on the set of real numbers.
$a R b$ if

1 + a b > 0

Find whether relation is reflexive, symmetric or transitive.

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Let $A$ be the set of all human beings in a town at a particular time. Determine whether of the following relation is reflexive, symmetric and transitive :

R = {(x, y) : x

and $y$ work at the same place

}

MEDIUM

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Let

n

be a fixed positive integer. Define a relation

R

Z

as follows:

(a, b) \in R \Leftrightarrow a - b

is divisible by

n

. Show that

R

is an equivalence relation on

Z

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

If $A = \{1, 2, 3, 4\}$ define relations on $A$ which have properties of being reflexive, transitive but not symmetric?

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Test whether the following relation

R_{1}

is reflexive, symmetric and transitive :

R_{1}

Q_{0}

defined by

(a, b) \in R_{1} \Leftrightarrow a = \frac{1}{b}

EASY

Physics>Modern Physics>Nuclei>Mass-energy and Nuclear Binding Energy

Let

A = {a, b, c}

and the relation $R$ be defined on $A$ as follows:

R = \{(a, a), (b, c), (a, b)\}

. Then, write minimum number of ordered pairs to be added in $R$ to make it reflexive and transitive.

Which agreement was signed between India and China in 1993 to maintain peace along the Line of Actual Control?

Important Questions on Nuclei

Learn and Prepare for any exam you want !