Relations Test 24

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Relations Test 24

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Weekly Quiz Competition

Question 1
1 / -0

If $$p - q > 0$$, which of the following is true?

A
If q = 0, then p < 0

B
If q < 0, then p < 0

C
If p > 0, then q > 0

D
If q = 0, then p > 0

E
If q < 1, then p > 1
Solution
- Given $$p-q>0$$
- If $$q=0$$ , then $$p-0>0$$ , which gives $$p>0$$
- Therefore option $$D$$ is correct
Question 2
1 / -0

Let $$R$$ be a relation defined on the set $$Z$$ of all integers and $$xRy$$ when $$x + 2y$$ is divisible by $$3$$. Then

A
$$R$$ is not transitive

B
$$R$$ is symmetric only

C
$$R$$ is an equivalence relation

D
$$R$$ is not an equivalence relation

Solution

$$aRa$$ is positive as $$a+2a=3a$$ is divisible by $$3$$
Hence reflexive
If $$aRb$$ is positive $$a+2b$$ is divisible by $$3$$
Hence $$3a+3b-(2a+b)$$ is divisible by $$3$$
$$\Rightarrow 2a+b$$ is divisible by $$3$$
ie $$aRb$$ is positive
$$\Rightarrow bRa$$ is positive hence symmetric
$$aRb,bRc$$ is $$a+2b,b+2c$$ divisible by $$3$$
$$\Rightarrow a+2b+b+2c$$ divisible by $$3$$
$$\Rightarrow a+3b+2c$$ divisible by $$3$$
So $$a+2c$$ divisible by $$3$$ hence transitive
So it is an equivalence relation.
Question 3
1 / -0

The relation R define on the set of natural numbers as {(a, b) : a differs from b by 3} is given.

A
$$\{(1, 4), (2, 5), (3, 6), ........\}$$

B
$$\{(4, 1), (5, 2), (6, 3), ........\}$$

C
$$\{(1, 3), (2, 6), (3, 9), ........\}$$

D
None of above

Solution

Let $$R=\{(a,b): a, b \in N, a-b=3\}$$
$$=\{(n+3,n):n \in N\}$$
$$=\{(4,1), (5, 2), (6, 3).......\}$$
Hence, option B is correct.
Question 4
1 / -0

The relation 'has the same father as' over the set of children is:

A
Only reflective

B
Only symmetric

C
Only transitive

D
An equivalence relation

Solution

Let this relation be called as $$R$$.
Then, if $$xRx$$ then clearly $$xRx$$ for all $$x$$. Also $$xRy$$ it means $$yRx$$
If $$xRy, yRz$$ it means $$x$$ has same father as $$y$$ and $$y$$ has same father as $$z$$.
So we can say that, $$x$$ has same father as $$z$$
Thus, $$xRz$$.
Thus, the given relation is reflexive, symmetric and transitive.
Hence, the relation is an equivalence relation.
Option D is correct.
Question 5
1 / -0

If A and B are two non-empty sets having n elements in common, then what is the number of common elements in the sets $$A\times B$$ and $$B\times A$$?

A
$$n$$

B
$$n^2$$

C
$$2n$$

D
Zero

Solution

Say A has x elements and B has y elements in total.

Their cartesian product $$A\times B$$ will have $$x\times y$$ elements.

Hence if they have n elements in common. $$n^2$$ common elements are present in the products $$A\times B$$ and $$B\times A$$
Question 6
1 / -0

Let $$A=\left\{ x\in W,the\quad set\quad of\quad whole\quad numbers\quad and\quad x<3 \right\} $$
$$B=\left\{ x\in N,the\quad set\quad of\quad natural\quad numbers\quad and\quad 2\le x<4 \right\} $$ and $$C=\left\{ 3,4 \right\} $$, then how many elements will $$\left( A\cup B \right) \times C$$ conatin?

A
$$6$$

B
$$8$$

C
$$10$$

D
$$12$$

Solution

$$A=\{0,1,2\}, B=\{2,3\}$$ and $$C=\{3,4\}$$
$$A \cup B=\{0,1,2,3\}$$
No. of elements in $$(A\cup B)\times C$$ $$=$$ No. of elements in $$(A \cup B)\times $$ No. of elements in $$C$$$$=4\times 2=8$$
Question 7
1 / -0

Let $$A = \left\{ a,b,c,d \right\}$$ and $$ B=\left\{ x,y,z \right\}$$. What is the number of elements in $$ A\times B$$?

A
$$6$$

B
$$7$$

C
$$12$$

D
$$64$$

Solution

Given sets are $$A=\{a,b,c,d\}$$ and $$\{x,y,z\}$$
So $$A\times B=\{(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z),(d,x),(d,y),(d,z)\}$$
No. of elements in $$A\times B$$ is $$3\times 4=12$$
Question 8
1 / -0

If $$A = \left\{ 1,2 \right\}$$, $$B = \left\{ 2,3 \right\}$$ and $$ C = \left\{ 3,4 \right\}$$, then what is the cardinality of $$ \left( A\times B \right) \cap \left( A\times C \right) $$

A
$$8$$

B
$$6$$

C
$$2$$

D
$$1$$

Solution

$$A = \left\{ 1,2 \right\}$$, $$B = \left\{ 2,3 \right\}$$ and $$ C = \left\{ 3,4 \right\}$$

Now, $$A\times B=\{ (1,2),(1,3),(2,2),(2,3)\}$$

$$A\times C=\{ (1,3),(1,4),(2,3),(2,4)\} $$ And

$$ (A\times B)\cap (A\times C)=\{ (1,3),(2,3)\} $$

So cardinality is $$2$$.

Hence, option C is correct.
Question 9
1 / -0

A and B are two sets having $$3$$ elements in common. If $$n(A)=5, n(B)=4$$, then what is $$n(A\times B)$$ equal to?

A
$$0$$

B
$$9$$

C
$$15$$

D
$$20$$

Solution

If $$n(A) =5$$ and $$n(B) = 4$$

For this type cases we know that the formula for the no. of elements in $$n(A\times B)$$ = $$5\times4 = 20$$
Question 10
1 / -0

Let $$Z$$ be the set of integers and $$aRb$$, where $$a, b\epsilon Z$$ if an only if $$(a - b)$$ is divisible by $$5$$.
Consider the following statements:
$$1.$$ The relation $$R$$ partitions $$Z$$ into five equivalent classes.
$$2.$$ Any two equivalent classes are either equal or disjoint.
Which of the above statements is/are correct?

A
$$1$$ only

B
$$2$$ only

C
Both $$1$$ and $$2$$

D
Neither $$1$$ nor $$2$$

Solution

A relation is defined on $$Z$$ such that $$aRb \Rightarrow (a-b)$$ is divisible by $$5$$,

For Reflexive: $$(a,a) \in R$$.
Since, $$(a-a) = 0$$ is divisible by $$5$$. Therefore, the realtion is reflexive.

For symmetric: If $$(a,b) \in R \Rightarrow (b,a) \in R$$.
$$(a,b) \in R \Rightarrow (a-b)$$ is divisible by $$5$$.
Now, $$(b-a) =-(a-b)$$ is also divisible by $$5$$. Therefore, $$(b,a) \in R$$
Hence, the relation is symmetric.

For Transitive: If $$(a,b) \in R $$ and $$ (b,a) \in R \Rightarrow (a,c) \in R$$.
$$(a,b) \in R \Rightarrow (a-b) $$ is divisible by $$5$$.
$$(b,c) \in R \Rightarrow (b-c)$$ is divisible by $$5$$.
Then $$(a-c)=(a-b+b-c)= (a-b)+(b-c) $$ is also divisible by $$5$$. Therefore, $$(a,c) \in R$$.
Hence, the relation is transitive.

There, the relation is equivalent.

Now, depending upon the remainder obtained when dividing $$(a-b)$$ by $$5$$ we can divide the set $$Z$$ iinto $$5$$ equivalent classes and they are disjoint i.e., there are no common elements between any two classes.

Therefore, the correct option is $$(C)$$.

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Relations Test 24

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Relations Test 24

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SHARING IS CARING

Question 1

If q = 0, then p < 0

If q < 0, then p < 0

If p > 0, then q > 0

If q = 0, then p > 0

If q < 1, then p > 1

Solution

Question 2

$$R$$ is not transitive

$$R$$ is symmetric only

$$R$$ is an equivalence relation

$$R$$ is not an equivalence relation

Solution

Question 3

$$\{(1, 4), (2, 5), (3, 6), ........\}$$

$$\{(4, 1), (5, 2), (6, 3), ........\}$$

$$\{(1, 3), (2, 6), (3, 9), ........\}$$

None of above

Solution

Question 4

Only reflective

Only symmetric

Only transitive

An equivalence relation

Solution

Question 5

$$n$$

$$n^2$$

$$2n$$

Zero

Solution

Question 6

$$6$$

$$8$$

$$10$$

$$12$$

Solution

Question 7

$$6$$

$$7$$

$$12$$

$$64$$

Solution

Question 8

$$8$$

$$6$$

$$2$$

$$1$$

Solution

Question 9

$$0$$

$$9$$

$$15$$

$$20$$

Solution

Question 10

$$1$$ only

$$2$$ only

Both $$1$$ and $$2$$

Neither $$1$$ nor $$2$$

Solution

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