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.

If Δ 1 = x s i n θ c o s θ - s i n θ - x 1 c o s θ 1 x and Δ 2 = x s i n 2 θ c o s 2 θ - s i n 2 θ - x 1 c o s 2 θ 1 x , x ≠ 0 ; then for all θ ∈ 0 , π 2 :

Δ 1 + Δ 2 = - 2 ( x 3 + x - 1 )

Δ 1 - Δ 2 = x ( c o s 2 θ - c o s 4 θ )

HARD

Earn 100

Which of the following is an example of Net Factor Income from Abroad (NFIA)?

50% studentsanswered this correctly

Important Questions on Solutions

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

Let

d \in R,

and

A = [\begin{matrix} - 2 & 4 + d & (\sin θ) - 2 \\ 1 & (\sin θ) + 2 & d \\ 5 & (2 \sin θ) - d & (- \sin θ) + 2 + 2 d \end{matrix}],

θ \in [0, 2 π] .

If the minimum value of

d e t (A)

8,

then a value of

d

is:

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

In series LCR circuit

R = 18 Ω

and impedance is

33 Ω

. An RMS voltage of

220 V

is applied across the circuit. The true power consumed in the AC circuit is

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

A small town with a demand of $1200 K W$ of electric power of $220 V$ is situated $20 K m$ away from an electric plant generating power at $440 V$ . The resistance of the two wireline carrying power is $0.5 Ω p e r k m .$ The town gets power from the line through a $4000 - 220 V$ step-down transformer at a sub-station in the town. Estimate the line power loss in the form of heat.

MEDIUM

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

|\begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix}| = a x - 12

, then

a

is equal to:

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

A = [\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}]

, then for all

θ \in (\frac{3 π}{4}, \frac{5 π}{4})

\det (A)

lies in the interval :

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

An alternating voltage

V = V_{0} \sin ω t

is applied across a circuit and as a result, a current

I = I_{0} \sin (ω t + \frac{π}{2})

flows in it. The power consumed per cycle is

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

The sum of the real roots of the equation

|\begin{matrix} x & - 6 & - 1 \\ 2 & - 3 x & x - 3 \\ - 3 & 2 x & x + 2 \end{matrix}| = 0,

is equal to:

HARD

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

S = \{x \in [0, 2 π] : |\begin{matrix} 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{matrix}| = 0\},

then

\sum_{x \in S} \tan (\frac{π}{3} + x)

is equal to:

MEDIUM

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

Show that the relation

R

R

is defined as

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

, is reflexive and transitive but not symmetric.

MEDIUM

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

Let the numbers

2, b, c

be in an A.P. and

A = [\begin{matrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{matrix}]

. If

d e t (A) \in [2,16],

then

c

lies in the interval:

HARD

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

Let

A = [\begin{matrix} 2 & b & 1 \\ b & b^{2} + 1 & b \\ 1 & b & 2 \end{matrix}],

where

b > 0

. Then the minimum value of

\frac{\det (A)}{b}

is:

MEDIUM

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

Δ_{1} = |\begin{matrix} x & s i n θ & c o s θ \\ - s i n θ & - x & 1 \\ c o s θ & 1 & x \end{matrix}|

and

Δ_{2} = |\begin{matrix} x & s i n 2 θ & c o s 2 θ \\ - s i n 2 θ & - x & 1 \\ c o s 2 θ & 1 & x \end{matrix}|,

x \neq 0;

then for all

θ \in (0, \frac{π}{2})

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

The inverse of matrix is

[\begin{matrix} 2 & 5 & 0 \\ 0 & 1 & 1 \\ - 1 & 0 & 3 \end{matrix}]

EASY

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

There are two values of

a

which makes determinant,

Δ = |\begin{matrix} 1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2 a \end{matrix}| = 86

, then sum of these number is

MEDIUM

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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

Chemistry>Physical Chemistry>Solutions>Van't Hoff's Factor

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 of the following is an example of Net Factor Income from Abroad (NFIA)?

Important Questions on Solutions

Learn and Prepare for any exam you want !