Relations Test 7

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Relations Test 7

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Weekly Quiz Competition

Question 1
1 / -0

Let $$R$$ be the relation on $$Z$$ defined by $$R = \{(a, b): a, b \in z, a - b$$ is an integer$$\}$$. Find the domain and Range of $$R$$.

A
$$z, z$$

B
$$z^+, z$$

C
$$z, z^-$$

D
None of these

Solution

Given:

$$R=\{ (a, b) : a, b, \in z, a-b \text{ is an integer}\}$$

As difference of integers are also integers so,

Domain of $$R = z$$

Range of $$R = z$$, as

$$a-b$$ spans the whole integer values.
Question 2
1 / -0

Find x and y, if $$(x+3,5)=(6,2x+y)$$.

A
x=3, y=-1

B
x=6, y=2

C
x=2,y=3

D
none of these

Solution

By the definition of equality of ordered pairs
$$(x+3,5)=(6,2x+y)$$
$$x+3=6$$ and $$5=2x+y$$
$$x=3$$ and $$5=2x+y$$
$$x=3$$ and $$5=6+y$$
$$x=3$$ and $$y=-1$$
Question 3
1 / -0

A relation $$R$$ is defined from $$\left\{ 2,3,4,5 \right\} $$ to $$\left\{ 3,6,7,10 \right\} $$ by:
$$xRy\Leftrightarrow x$$ is relatively prime to $$y$$. Then, domain of $$R$$ is

A
$$\left\{ 2,3,5 \right\} $$

B
$$\left\{ 3,5 \right\} $$

C
$$\left\{ 2,3,4 \right\} $$

D
$$\left\{ 2,3,4,5 \right\} $$

Solution

Let's first write R as a set of ordered pairs,

$$R=\{(2,3),(2,7),(3,7),(3,10),(4,3),(4,7),(5,3),(5,6),(5,7)\}$$

So, Domain of R is $$\{2,3,4,5\}$$
Question 4
1 / -0

The relation $$R$$ in $$N\times N$$ such that $$(a,b)R(c,d)\Leftrightarrow a+d=b+c$$ is

A
reflexive but not symmetric

B
reflexive and transitive but not symmetric

C
an equivalence relation

D
none of these

Solution

If (a,b)R(c,d) then (c,d)R(a,b) as if a+d=b+c then b+d=a+c .
Thus relation R is symmetric .

For some a,b in N , (a,b)R(a,b) as a+b=b+a.
Thus every element of $$N\times N$$ is related to itself,
So relation R is also reflexive .

For some a,b,c,d,e,f in N, if (a,b)R(c,d) and (c,d)R(e,f) then (a,b)R(e,f),
As if a+d=b+c...1 and c+f=d+e….2 then a+f=b+e [since eq.1- eq.2 gives a-e=b-f$$\rightarrow$$a+f=b+e ]
So, R is also symmetric relation .

Thus R is a equivalence relation.So, C is correct answer .
Question 5
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Let $$R$$ be the relation over the set of all straight lines in a plane such that $${l}_{1}$$ $$R$$ $${l}_{2}\Leftrightarrow {l}_{1}\bot {l}_{2}$$. Then, $$R$$ is

A
symmetric

B
reflexive

C
transitive

D
an equivalence relation

Solution

For some lines $$l_1,l_2$$, We know that $$l_1\perp l_2 \leftrightarrow l_2\perp l_1$$
So, given relation is symmetric .

For some lines $$l_1,l_2,l_3$$, We know that if $$l_1\perp l_2 \ and \ l_2\perp l_3$$, then $$l_1\parallel l_3$$
So, given relation is not transitive .

For some line $$l$$, We know that it is never possible that $$l\perp l$$.
So, given relation is not reflexive .

Thus $$A$$ is correct answer .
Question 6
1 / -0

Let $$R$$ be a relation on the set $$N$$ given by $$R=\left\{ \left( a,b \right) :a=b-2,b>6 \right\}$$. Then

A
$$(2,4)\in R$$

B
$$(3,8)\in R$$

C
$$(6,8)\in R$$

D
$$(8,7)\in R$$

Solution

$$a=b-2,b>6$$

Above relation also imply $$a>6-2\rightarrow a>4$$

Among the options

$$C$$ is correct answer as $$6>4$$ and aslo $$6=8-2$$ .
Question 7
1 / -0

If $$A=\left\{ a,b,c \right\} $$, then the relation $$R=\left\{ \left( b,c \right) \right\} $$ on $$A$$ is

A
reflexive only

B
symmetric only

C
transitive only

D
reflexive and transitive only

Solution

The relation R contains (b,c) but do not contain (c,b) , so it is not symmetric .
The relation R do not relates any element of set A with itself, so it is not reflexive .
The relation R contains only one pair (b,c), so it is transitive .

Thus option C is correct .
Question 8
1 / -0

Which of the following is not an equivalence relation on $$Z$$?

A
$$aRb\Leftrightarrow a+b$$ is an even integer

B
$$aRb\Leftrightarrow a-b$$ is an even integer

C
$$aRb\Leftrightarrow a< b$$

D
$$aRb\Leftrightarrow a= b$$

Solution

Among the Options

$$Opt:[C]$$

$$aRb\Leftrightarrow a< b$$

As, $$a<b$$ doesn't imply $$b<a$$ ,which make the relation asymmetric and thus it is not an equivalence relation .
Question 9
1 / -0

If $$A=\left\{ 1,2,3 \right\} , B=\left\{ 1,4,6,9 \right\} $$ and $$R$$ is a relation from $$A$$ to $$B$$ defined by $$x$$ is greater than $$y$$. The range of $$R$$ is

A
$$\left\{ 1,4,6,9 \right\} $$

B
$$\left\{ 4,6,9 \right\} $$

C
$$\left\{ 1 \right\} $$

D
none of these

Solution

First Let's write R as set of ordered pairs,
$$R=\{(2,1),(3,1)\}$$
We can simply see from set that range of $$R = \{1\}$$
Question 10
1 / -0

A relation $$\phi$$ from $$C$$ to $$R$$ is defined by $$x\phi y\Leftrightarrow \left| x \right| =y$$. Which one is correct?

A
$$(2+3i)\phi 13$$

B
$$3\phi (-3)$$

C
$$(1+i)\phi 2$$

D
$$i\phi 1$$

Solution

Given, a relation $$\phi$$ from $$C$$ to $$R$$ is defined by $$x\phi y\Leftrightarrow \left| x \right| =y$$.

Now, $$|(2+3i)|=\sqrt{2^2+3^2}=\sqrt{13}$$.

So $$(2+3i)$$ and $$13$$ are not related.

Again, since $$y>0$$ this gives $$3$$ not related to $$-3$$.

$$|1+i|=\sqrt{1+1}=\sqrt 2$$

So $$1+i$$ and $$2$$ are not related.

Also $$|i|=1$$ so $$i\phi 1$$ holds.

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Relations Test 7

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Relations Test 7

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Question 1

$$z, z$$

$$z^+, z$$

$$z, z^-$$

None of these

Solution

Question 2

x=3, y=-1

x=6, y=2

x=2,y=3

none of these

Solution

Question 3

$$\left\{ 2,3,5 \right\} $$

$$\left\{ 3,5 \right\} $$

$$\left\{ 2,3,4 \right\} $$

$$\left\{ 2,3,4,5 \right\} $$

Solution

Question 4

reflexive but not symmetric

reflexive and transitive but not symmetric

an equivalence relation

none of these

Solution

Question 5

symmetric

reflexive

transitive

an equivalence relation

Solution

Question 6

$$(2,4)\in R$$

$$(3,8)\in R$$

$$(6,8)\in R$$

$$(8,7)\in R$$

Solution

Question 7

reflexive only

symmetric only

transitive only

reflexive and transitive only

Solution

Question 8

$$aRb\Leftrightarrow a+b$$ is an even integer

$$aRb\Leftrightarrow a-b$$ is an even integer

$$aRb\Leftrightarrow a< b$$

$$aRb\Leftrightarrow a= b$$

Solution

Question 9

$$\left\{ 1,4,6,9 \right\} $$

$$\left\{ 4,6,9 \right\} $$

$$\left\{ 1 \right\} $$

none of these

Solution

Question 10

$$(2+3i)\phi 13$$

$$3\phi (-3)$$

$$(1+i)\phi 2$$

$$i\phi 1$$

Solution

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