Question 10

# 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.

(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.

Answer

**(i)** Let *A* = {5, 6, 7}.

Define a relation R on* A* as R = {(5, 6), (6, 5)}.

Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R.

Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric.

=> (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R

∴R is not transitive.

Hence, relation R is symmetric but not reflexive or transitive.

**(ii) **Consider a relation R in **R **defined as:

R = {(*a*, *b*): *a* < *b*}

For any *a *∈ R, we have (*a*, *a*) ∉ R since *a* cannot be strictly less than *a* itself. In fact, *a* = *a*.

**∴ **R is not reflexive.

Now,

(1, 2) ∈ R (as 1 < 2)

But, 2 is not less than 1.

**∴ **(2, 1) ∉ R

**∴ **R is not symmetric.

Now, let (*a*, *b*), (*b*, *c*) ∈ R.

⇒ *a* < *b* and *b* < *c*

⇒ *a* < *c*

⇒ (*a*, *c*) ∈ R

**∴ **R is transitive.

Hence, relation R is transitive but not reflexive and symmetric.

**(iii)** Let *A* = {4, 6, 8}.

Define a relation R on A as:

*A* = {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)}

Relation R is reflexive since for every *a* ∈ *A*, (*a*, *a*) ∈R i.e., (4, 4), (6, 6), (8, 8)} ∈ R.

Relation R is symmetric since (*a*, *b*) ∈ R ⇒ (*b*, *a*) ∈ R for all *a*, *b* ∈ R.

Relation R is not transitive since (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R.

Hence, relation R is reflexive and symmetric but not transitive.

**(iv)** Define a relation R in **R** as:

R = {*a*, *b*): *a*^{3} ≥ *b*^{3}}

Clearly (*a*, *a*) ∈ R as *a*^{3} = *a*^{3}.

**∴ **R is reflexive.

Now,

(2, 1) ∈ R (as 2^{3} ≥ 1^{3})

But,

(1, 2) ∉ R (as 1^{3} < 2^{3})

**∴** R is not symmetric.

Now,

Let (*a*, *b*), (*b*, *c*) ∈ R.

⇒ *a*^{3} ≥ *b*^{3} and *b*^{3} ≥ *c*^{3}

⇒ *a*^{3} ≥ *c*^{3}

⇒ (*a*, *c*) ∈ R

**∴ **R is transitive.

Hence, relation R is reflexive and transitive but not symmetric.

**(v) ** Let *A* = {−5, −6}.

Define a relation R on *A* as:

R = {(−5, −6), (−6, −5), (−5, −5)}

Relation R is not reflexive as (−6, −6) ∉ R.

Relation R is symmetric as (−5, −6) ∈ R and (−6, −5}∈R.

It is seen that (−5, −6), (−6, −5) ∈ R. Also, (−5, −5) ∈ R.

**∴ **The relation R is transitive.

Hence, relation R is 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:- 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:- 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.
- 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:- If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
- Q:-
Find the direction cosines of a line which makes equal angles with the coordinate axes.

- Q:-
Show that the function

*f*: R_{*}→ R_{*}defined by*f(x)*= 1/x is one-one and onto,where R_{*}is the set of all non-zero real numbers. Is the result true, if the domain R_{*}is replaced by N with co-domain being same as R_{* }? - Q:-
Answer the following as true or false.

\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

- Q:-
Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

- Q:- .">
Consider

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

with**.** - 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 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:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
- Q:-
A particle moves along the curve 6y = x

^{3}+ 2. Find the points on the curve at which the*y*-coordinate is changing 8 times as fast as the*x*-coordinate.

callboy
2019-08-23 02:10:20

in part v set is trans. then (-6,-5) & (-5,-6) both are in relation

angshika
2019-08-21 09:28:45

Thanks for the help

Kajol
2018-12-23 12:01:37

In v. If -6,-6 belongs to R then it will be reflexive (a,a) belongs to R therefore v answer is correct

Sunny
2018-07-15 21:04:10

Try to improve much more

Sachin
2015-04-17 13:33:47

I think, it is correct because (-6,-6) does not belongs to relation set R. Properties of Relation is A realtion R on set A is reflexive if aRa for all a belongs to A i.e. is (a,a) belongs to R for all a belongs to R => each element a of A is related to itself. Ex: Let A = {a,b} and R = {(a,a),(a,b),(b,a)} then R is reflexive as aRa belongs to R but it is not reflexive for pair (b,b) does not belongs to R.

imer
2015-04-16 11:55:15

plz check part v it does not seems correct as -6,-6 doesnot belongs to R

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