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Question 6

## Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Answer

Let *A* = {1, 2, 3}.

A relation R on *A* is defined as R = {(1, 2), (2, 1)}.

It is seen that (1, 1), (2, 2), (3, 3) ∉R.

∴ R is not reflexive.

Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.

Now, (1, 2) and (2, 1) ∈ R

However,

(1, 1) ∉ R

∴ R is not transitive.

Hence, R is symmetric but neither reflexive nor transitive.

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(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 exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- 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. - Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:-
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(i)

*f*:**R → R**defined by*f(x)*= 3 – 4x(ii)

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- Q:-
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*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
- Q:-
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*f*: R → R be defined as f(x) = 3x. Choose the correct answer.(A)

*f*is one-one onto(B)

*f*is many-one onto(C)

*f*is one-one but not onto(D)

*f*is neither one-one nor onto.

- Q:-
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- Q:-
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*x*+ 4*y*Subject to the constraints:

*x*+*y*≤ 4,*x*≥ 0,*y*≥ 0 - Q:-
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\begin{vmatrix} \mathbf{2} & \mathbf{4} \\ \mathbf{-5} & \mathbf{-1} \end{vmatrix} - Q:- Find the rate of change of the area of a circle with respect to its radius r when

(a) r = 3 cm

(b) r = 4 cm - Q:-
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).

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- Q:- \begin{align} \int sec x . \left(sec x + tan x\right) .dx \end{align}
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The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

- Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:-
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

- Q:- \begin{align} \int \frac {sec^2 x}{Coses^2 x} . dx\end{align}
- Q:-
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

- Q:- Find the principal value of \begin{align} cosec^{-1}\left({-\sqrt2}\right)\end{align}
- Q:- If \begin{align} \frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}\end{align} such that f(2) = 0 , then f(x) is

\begin{align} (A) x^4 + \frac {1}{x^3} - \frac{129}{8} \;\;\;\;(B) x^3 + \frac{1}{x^4} + \frac{129}{8}\end{align}

\begin{align} (c) x^3 + \frac {1}{x^4} + \frac{129}{8} \;\;\;\;(D) x^3 + \frac{1}{x^4} - \frac{129}{8}\end{align} - Q:-
Determine order and degree(if defined) of differential equation y

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