Relations Test 33

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Relations Test 33

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Question 1
1 / -0

Let A and B be two sets such that $$A\times B=\left\{ \left( a,1 \right) ,\left( b,3 \right) ,\left( a,3 \right) ,\left( b,1 \right) ,\left( a,2 \right) ,\left( b,2 \right) \right\} ,$$ then

A
$$A=\left\{ 1,2,3 \right\} $$ and $$B=\left\{ a,b \right\} $$

B
$$A=\left\{ a,b \right\} $$ and$$ B=\left\{ 1,2,3 \right\} $$

C
$$A=\left\{ 1,2,3 \right\} $$ and $$B\subset \left\{ a,b \right\} $$

D
$$A\subset \left\{ a,b \right\} $$ and $$B\subset \left\{ 1,2,3 \right\} $$

Solution

$$A$$ is the first element in the cartesian product $$A\times B=\left\{a\,,b\,,a\,,b\,,a\,,b\right\}$$
and $$B$$ is the second element in the cartesian product $$A\times B=\left\{1,\,3\,,3\,,1\,,2\,,2\right\}$$
$$\therefore$$ elements of $$A=\left\{a,b\right\}$$ and $$B=\left\{1\,,2\,,3\right\}$$
Question 2
1 / -0

A relation R is defined from {2,3,4,5} to {3,6,7,10} by XRY $$\Leftrightarrow $$ X is relatively prime to Y, then domain of R is

A
{2,3,5}

B
{3,5}

C
{2,5}

D
{2,3,4,5}

Solution

Given:
From $$\left\{2,3,4,5\right\}$$ to $$\left\{3,6,7,10\right\}$$

$$x$$ is related to $$y$$ iff $$x$$ is relatively prime to $$y$$

$$2$$ is relatively prime to $$3,7$$

$$3$$ is relatively prime to $$7,10$$

$$4$$ is relatively prime to $$3,7$$

$$5$$ is relatively prime to $$3,6,7$$

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

If the cardinality of a set $$A$$ is $$4$$ and that of a set $$B$$ is $$3$$, then what is the cardinality of the set $$A\Delta B$$.

A
$$1$$

B
$$5$$

C
$$7$$

D
$$Cannot\ be\ determined$$

Solution

$$n(A\Delta B)=n(A)+n(B)-n(A\cap B)$$
Here we don`t know the $$n(A\cap B)$$
So the answer cannot be Determined.
Question 4
1 / -0

What type of a relation is "Less than" in the set of real numbers?

A
only symmertric

B
only transitive

C
only reflexive

D
equivalence relation

Solution

If $$a\in\,R,\,a$$ is not less than $$a$$.
$$\Rightarrow\,$$ The relation “less than” is not reflexive.
If $$a,b\in R,$$ and $$a$$ is less than $$b$$ then $$b$$ is not less than $$a$$.
$$\Rightarrow\,$$ The relation “less than” is not symmetric.
If $$a,b,c\in\,R,$$ and $$a$$ is less than $$b$$ and $$b$$ is less than $$c$$ then $$a$$ is less than $$c$$.
$$\Rightarrow\,$$The relation “less than” is transitive.
$$\Rightarrow\,$$The relation “less than” is not an equivalence relation.
Hence option$$(b)$$ is correct.
Question 5
1 / -0

For real numbers x and y, define xRy if $$x-y+\sqrt{2}$$ is an irrational number. Then the relation R is?

A
Not Reflexive

B
Not Symmetric

C
Not Transitive

D
None of these

Solution
Question 6
1 / -0

Let $$A$$ be set of first ten natural numbers and $$R$$ be a relation on A, defined by $$(x,y)\in R\Rightarrow x+2y=10$$, then domain of $$R$$ is

A
{1, 2, 3, ......,10}

B
{2, 4, 6, 8}

C
{1, 2, 3, 4}

D
{2, 4, 6, 8, 10}
Question 7
1 / -0

Let $$S=\{1, 2, 3, 4, 5\}$$ and let A$$=S\times S$$. Defined the relation on the R on A as follows (a, b)R(c, d) if an only if ad$$=$$bc. Then R is?

A
Reflexive only

B
Symmetric only

C
Transitive only

D
An equivalence

Solution
Question 8
1 / -0

Let $$g(x)-f(x)=1$$. If $$f(x)+f(1-x)=2\forall x\in R$$, then $$g(x)$$ of symmetrical about?

A
The origin

B
The line $$x=\dfrac{1}{2}$$

C
The point $$(1, 0)$$

D
The point $$\left(\dfrac{1}{2}, 0\right)$$

Solution
Question 9
1 / -0

A is a set having $$6$$ distinct elements. The number of distinct function from A to A which are not bijections is?

A
$$6!-6$$

B
$$6^6-6$$

C
$$6^6-6!$$

D
$$6!$$

Solution

Total$$-$$ bijections

$$6^6-6!$$.
Question 10
1 / -0

N is the set of natural numbers. The relation R is defined on the N$$\times$$N as follows $$_{a\cdot b}R_{c\cdot d}\Leftrightarrow a+d=b+c$$. Then, R is?

A
Reflexive only

B
Symmetric only

C
Transition only

D
An equivalence

Solution

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Relations Test 33

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Relations Test 33

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Question 1

$$A=\left\{ 1,2,3 \right\} $$ and $$B=\left\{ a,b \right\} $$

$$A=\left\{ a,b \right\} $$ and$$ B=\left\{ 1,2,3 \right\} $$

$$A=\left\{ 1,2,3 \right\} $$ and $$B\subset \left\{ a,b \right\} $$

$$A\subset \left\{ a,b \right\} $$ and $$B\subset \left\{ 1,2,3 \right\} $$

Solution

Question 2

{2,3,5}

{3,5}

{2,5}

{2,3,4,5}

Solution

Question 3

$$1$$

$$5$$

$$7$$

$$Cannot\ be\ determined$$

Solution

Question 4

only symmertric

only transitive

only reflexive

equivalence relation

Solution

Question 5

Not Reflexive

Not Symmetric

Not Transitive

None of these

Solution

Question 6

{1, 2, 3, ......,10}

{2, 4, 6, 8}

{1, 2, 3, 4}

{2, 4, 6, 8, 10}

Question 7

Reflexive only

Symmetric only

Transitive only

An equivalence

Solution

Question 8

The origin

The line $$x=\dfrac{1}{2}$$

The point $$(1, 0)$$

The point $$\left(\dfrac{1}{2}, 0\right)$$

Solution

Question 9

$$6!-6$$

$$6^6-6$$

$$6^6-6!$$

$$6!$$

Solution

Question 10

Reflexive only

Symmetric only

Transition only

An equivalence

Solution

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