Relations Test 20

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Relations Test 20

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Weekly Quiz Competition

Question 1
1 / -0

Write the properties that the relation "is greter that" satisfies in the set of all positive integers

A
Reflexive

B
Symmetric

C
Antisymmetric

D
Transitive

Solution

If in a relation, an element $$ a $$ is related to $$ b $$, and $$ b $$ is related to a third element $$ c $$, the relation $$ R $$ is said to be transitive if $$ a $$ is also related to $$ c $$

Clearly, when $$ a > b $$ and $$ b > c $$ then $$ a $$ $$ > c $$
Hence, the relation is transitive.
Question 2
1 / -0

If $$\displaystyle R=R^{-1}$$ then the relation R is ________

A
reflexive

B
symmetric

C
anti-symmetric

D
transitive

Solution

When a relation is equal to its inverse, then the relation is symmetric.
Question 3
1 / -0

The domain of the relation R = $$\displaystyle \left \{ \left ( x,y \right ):x,y\epsilon N \ and\ x+y\leq 3 \right \}$$ is____

A
{1,2,3}

B
{1,2}

C
{...-1,0,1,2,3}

D
None of these

Solution

The only natrual numbers, less than equal to $$ 3 $$ are $$ 1, 2, 3 $$

From these, only the sum of $$ 1 ,2 $$ gives $$ 3 $$

So the domain of he relation is {$$ 1,2$$}
Question 4
1 / -0

If $$\displaystyle n\left ( A\times B \right )=36$$ then n(A) can possibly be____

A
$$7$$

B
$$8$$

C
$$9$$

D
$$10$$

Solution

$$n(A\times B)=n(A)\times n(B)$$

Hence $$n(A)$$ must be a factor of $$36$$. only possible answer is $$B:9$$
Question 5
1 / -0

The relation 'is a sister of' in the set of human beings is____

A
only transitive

B
only symmetric

C
equivalent

D
None of these

Solution

A relation is said to be transitive when an element $$ a $$ is related to $$ b $$ and element $$ b $$ is related to $$ c $$, then $$ c $$ is also related to $$ a $$

We know that if $$ a $$ is the sister of $$ b $$ and $$ b $$ is sister of $$ c $$, then $$ c $$ is also a sister of $$ a $$

Thus, the relation " is a sister of " is a transitive relation.
Question 6
1 / -0

The relation 'is a factor of' on the set of natural numbers is not___________

A
reflective

B
symmetric

C
anti symmetric

D
transitive

Solution

A relation is said to be symmetric if an element a is related to b then b is also related to a.

Suppose factor of a number say $$ 10 = 1,2,5,10 $$
We can notice that factors of $$ 10 $$ are all these numbers, but the reverse is not true for each factor. That is $$ 10 $$ is not a factor of $$ 1 $$ or $$ 2 $$ or $$ 5 $$.

So, this relation is not symmetric.
Question 7
1 / -0

If $$\displaystyle n\left ( P\times Q \right )=0$$ then n(P) can possibly be

A
0

B
10

C
20

D
Any value

Solution

$$ n(P) $$ can be of any value as we are not sure of $$ n(Q) $$
Hence, $$ n(P) $$ can take any of the given values.
Question 8
1 / -0

If a R b, b R c and a R c, and a,b,c,$$\displaystyle \epsilon $$ A, then the relation R on the set A is said to be a/an_____ relation

A
reflexive

B
symmetric

C
transitive

D
equivalence

Solution

If in a relation, an element $$ a $$ is related to $$ b $$, and $$ b $$ is related to a third element $$ c $$, the relation $$ R $$ is said to be transitive if $$ a $$ is also related to $$ c $$
Question 9
1 / -0

Let $$R$$ be the relation in the set $$\left\{1, 2, 3, 4\right\}$$ given by $$R=\left\{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\right\}$$. Choose the correct answer.

A
$$R$$ is reflexive and symmetric but not transitive.

B
$$R$$ is reflexive and transitive but not symmetric.

C
$$R$$ is symmetric and transitive but not reflexive.

D
$$R$$ is an equivalence relation.

Solution

$$R=\left\{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\right\}$$
It is seen that $$(a, a)\in R$$, for every $$a\in \left\{1, 2, 3, 4\right\}$$.
$$\therefore $$ $$R$$ is reflexive.
It is seen that $$(1, 2)\in R$$, but $$(2, 1)\notin R$$.
$$\therefore$$ $$R$$ is not symmetric.
Also, it is observed that $$(a, b), (b, c)\in R\Rightarrow (a, c)\in R$$ for all $$a, b, c \in \left\{1, 2, 3, 4\right\}$$
$$\therefore$$ $$R$$ is transitive.
Hence,$$R$$ is reflexive and transitive but not symmetric.
The correct answer is B.
Question 10
1 / -0

If $$A=\left \{1, 2,3\right \}$$ and $$B=\left \{3,8\right \}$$, then $$(A\cup B)\times (A\cap B)$$ is equal to

A
$$\left \{(8,3), (8,2), (8,1), (8,8)\right \}$$

B
$$\left \{(1,2), (2,2), (3,3), (8,8)\right \}$$

C
$$\left \{(3,1), (3,2), (3,3), (3,8)\right \}$$

D
$$\left \{(1,3), (2,3), (3,3), (8,3)\right \}$$

Solution

Given, $$A=\left \{1, 2,3\right \}$$ and $$B=\left \{3,8\right \}$$,
Therefore, $$A\cup B=\left \{1,2,3\right \}\cup \left \{3,8\right \}=\left \{1,2,3,8\right \}$$
and $$A\cap B=\left \{1,2,3\right \}\cap \left \{3,8\right \}=\left \{3\right \}$$
$$\therefore (A\cup B)\times (A\cap B)=\left \{1,2,3,8\right \}\times \left \{3\right \}$$
$$=\left \{(1,3), (2,3), (3,3), (8,3)\right \}$$

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Re-Attempt Weekly Quiz Competition

Relations Test 20

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Relations Test 20

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SHARING IS CARING

Question 1

Reflexive

Symmetric

Antisymmetric

Transitive

Solution

Question 2

reflexive

symmetric

anti-symmetric

transitive

Solution

Question 3

{1,2,3}

{1,2}

{...-1,0,1,2,3}

None of these

Solution

Question 4

$$7$$

$$8$$

$$9$$

$$10$$

Solution

Question 5

only transitive

only symmetric

equivalent

None of these

Solution

Question 6

reflective

symmetric

anti symmetric

transitive

Solution

Question 7

0

10

20

Any value

Solution

Question 8

reflexive

symmetric

transitive

equivalence

Solution

Question 9

$$R$$ is reflexive and symmetric but not transitive.

$$R$$ is reflexive and transitive but not symmetric.

$$R$$ is symmetric and transitive but not reflexive.

$$R$$ is an equivalence relation.

Solution

Question 10

$$\left \{(8,3), (8,2), (8,1), (8,8)\right \}$$

$$\left \{(1,2), (2,2), (3,3), (8,8)\right \}$$

$$\left \{(3,1), (3,2), (3,3), (3,8)\right \}$$

$$\left \{(1,3), (2,3), (3,3), (8,3)\right \}$$

Solution

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