Relations Test 38

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Relations Test 38

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Question 1
1 / -0

Let R be a relation on the set of integers given by $$ aRb \leftrightarrow a = 2^k .b $$ for some integer $$k.$$ Then $$R$$ is

A
an equivalence relation

B
reflexive and transitive but not symmetric

C
reflexive and transitive but not transitive

D
symmetric and transitive but nor reflexive

Solution
Question 2
1 / -0

Total number of equivalence relations defined in the set $$S=\left\{a, b, c\right\}$$ is?

A
$$5$$

B
$$3!$$

C
$$2^3$$

D
$$3^3$$

Solution
Question 3
1 / -0

Let $$A={1,2,3}$$. Then number of equivalence relations containing $$(1,2)$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

$$A=\left\{1,2,3\right\}$$
For equivalence relation containing $$(1,2)$$
For symmetric, it must consists $$(1,2)$$ and $$(2,1)$$.
For transitivity, it must consists $$(1,3)$$ and $$(3,2)$$ and $$(1,2),(2,1),(2,3),(3,1)$$
For reflexivity, it must consists $$(1,1)$$ and $$(2,2),(3,3)$$
$$\Rightarrow R=\left\{(1,1),(2,2),(2,3),(3,3),(1,2)(1,3),(2,1),(3,1),(3,1)\right\}$$
$$\Rightarrow$$ Only $$1$$ such relation is possible.
Question 4
1 / -0

If R is a reflexive relation on a finite set A having n-elements, and there are m ordered pairs in R. then?

A
$$m \geq n$$

B
$$m\leq n$$

C
$$m = n$$

D
$$m\neq n$$

Solution
Question 5
1 / -0

Let R be a relation from $$ A=\{1,2,3,4\}$$ to $$B=\{1,3,5\}$$such that R=[(a,b):a<b where a\in{A} and b\in{B}].What is $$RoR^{-1} $$ equal to

A
$$(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)$$

B
$$(3,1),(5,1),(3,2)(5,2),(5,3),(5,4)$$

C
$$(3,3),(3,5),(5,3),(5,5)$$

D
$$(3,3),(3,4),(4,5)$$

Solution
Question 6
1 / -0

Let $$R$$ a relation on the set $$N$$ be defined by $$\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$$. Then $$R$$ is

A
Reflexive

B
Symmetric

C
Transitive

D
None of these

Solution

Given R:{(x,y):x∈Ny∈N,2x+y=41}R:{(x,y):x∈Ny∈N,2x+y=41}

2x+y=412x+y=41

y=41−2xy=41−2x

Since y∈Ny>0y∈Ny>0

41−2x>041−2x>0

−2x>−41−2x>−41

2x<412x<41

x<412x<412

Since x∈N.x∈N. x can take values ={1,2,3....20}={1,2,3....20}

Domain of R={1,2,3.....20}R={1,2,3.....20}

Range of R is y=41−2xforx={1,2,3....20}y=41−2xforx={1,2,3....20}

Range of R={1,3,.......,37,39}R={1,3,.......,37,39}

When x=1y=41−2=39x=1y=41−2=39

When x=2y=41−2x×2=39x=2y=41−2x×2=39

When x=20y=40−2×20=1x=20y=40−2×20=1

Domain of f={1,2.....20}f={1,2.....20}

Range of f={1,3......37,39}f={1,3......37,39}

R is not reflexive

Since (1,1) does not satisfy

2x+y=412x+y=41

2×1+1≠412×1+1≠41

R is not symmetric since (1,39)∈R(1,39)∈R satisfies 2x+y=41but(39,1)∉R2x+y=41but(39,1)∉R

Since 2×39+1≠412×39+1≠41

but 2×1+39=412×1+39=41

Solution:R is not transitive since (1,39) ∈∈ but no values for x=39. satisfies the given relation
Question 7
1 / -0

$$f:\left( { - \infty ,\infty } \right) \to \left( {0,1} \right]$$ defined by $$f\left( x \right) = \frac{1}{{{x^2} + 1}}$$ is

A
one -one but not onto

B
onto but not one-one

C
bijective

D
neither one-one nor onto

Solution
Question 8
1 / -0

Let $$X=\left\{ x \epsilon {R};\cos(\sin\ x)=\sin(\cos\ x)\right\}$$. The number of elements in $$X$$ is

A
$$0$$

B
$$2$$

C
$$4$$

D
$$ not\ finite$$

Solution
Question 9
1 / -0

If the relation $$R:A \to B$$ where $$A = \left\{ {1,2,3,4} \right\},B = \left\{ {1,3,5} \right\}$$ is defined by $$R = \left\{ {\left( {x,y} \right):x < y,x \in A,y \in B} \right\}$$ then $$Ro{R^{ - 1}} = $$

A
$$\left\{ {\left( {1,3} \right),\left( {1,5} \right),\left( {2,3} \right),\left( {2,5} \right),\left( {3,5} \right),\left( {4,5} \right)} \right\}$$

B
$$\left\{ {\left( {3,1} \right),\left( {5,1} \right),\left( {3,2} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right)} \right\}$$

C
$$\left\{ {\left( {3,3} \right),\left( {3,5} \right),\left( {5,3} \right),\left( {5,5} \right)} \right\}$$

D
$$\left\{ {\left( {3,3} \right).\left( {3,4} \right),\left( {4,5} \right)} \right\}$$

Solution
Question 10
1 / -0

For real number $$x$$ and $$y$$, define a relationship $$R$$ {;$$x\ Ry$$ if and only if $$x-y+\sqrt{2}$$ is an irrational number }. Then the relation $$R$$ is

A
Reflexive

B
symmetric

C
transitive

D
an equivalence relation

Solution