Question 14

# 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.

Answer

R = {(*L*_{1}, *L*_{2}): L_{1} is parallel to *L*_{2}}

R is reflexive as any line *L*_{1} is parallel to itself i.e., (*L*_{1}, *L*_{1}) ∈ R.

Now,

Let (*L*_{1}, *L*_{2}) ∈ R.

⇒ *L*_{1} is parallel to *L*_{2.}

⇒ *L*_{2} is parallel to *L*_{1.}

⇒ (*L*_{2}, *L*_{1}) ∈ R

∴ R is symmetric.

Now,

Let (*L*_{1}, *L*_{2}), (*L*_{2}, *L*_{3}) ∈R.

⇒ *L*_{1} is parallel to *L*_{2}. Also, *L*_{2} is parallel to *L*_{3.}

⇒ *L*_{1} is parallel to *L*_{3.}

∴R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line *y* = 2*x* + 4 is the set of all lines that are parallel to the line *y* = 2*x* + 4.

Slope of line *y* = 2*x* + 4 is *m* = 2

It is known that parallel lines have the same slopes.

The line parallel to the given line is of the form *y* = 2*x* + *c*, where *c* ∈**R**.

Hence, the set of all lines related to the given line is given by *y* = 2*x* + *c*, where *c* ∈ **R**.

- 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:- 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:- Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by

(i) R = { (a,b) : |a - b| is a multiple of 4}

(ii) R = {(a,b):a = b} is an equivalence relation.

Find the set of all elements related to 1 in each case. - Q:-
Check the injectivity and surjectivity of the following functions:

(i)

*f*:**N → N**given by*f(x*) = x^{2}(ii)

*f*:**Z → Z**given by*f(x)*= x^{2}(iii)

*f*:**R → R**given by*f(x)*= x^{2}(iv)

*f*:**N → N**given by*f(x)*= x^{3}(v)

*f*:**Z → Z**given by*f(x)*= x^{3 } - 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:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- 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:- If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
- Q:-
Show that the Modulus Function

*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:-
Represent graphically a displacement of 40 km, 30° east of north.

- 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:- Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
- Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
- Q:-
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

- Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- .">
Consider

*f*: R_{+}→ [– 5, ∞) given by*f(x)*= 9x^{2}+ 6x – 5. Show that*f*is invertible

with**.** - Q:- . Is f one-one and onto? Justify your answer.

">

Let A = R – {3} and B = R – {1}. Consider the function *f* : A → B defined by

. Is f one-one and onto? Justify your answer.

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

- NCERT Chapter

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