Relations Test 11

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Relations Test 11

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Weekly Quiz Competition

Question 1
1 / -0

If $$A=\{b,c,d\}$$ and $$B=\{x,y\}$$. Find which of the following are elements of $$A \times A$$.

A
$$\{b,b\}$$

B
$$\{b,c\}$$

C
$$\{b,d\}$$

D
All of the above

Solution

$$A\times A \Rightarrow$$ the first element will be from $$A$$ and the second element will also be from $$A$$.
$$A \times A = \left\{\{b,b\}, \{b,c\}, \{b,d\},\{c,b\},\{c,c\},\{c,d\},\{d,b\},\{d,c\},\{d,d\}\right\}$$
Thus, all the options $$A,B$$ and $$C$$ are the elements of $$A \times A$$
Question 2
1 / -0

Suppose $$S=\{1,2\}$$ and $$T=\{a,b\}$$ then $$T \times S$$

A
$$(a,1),(a,2),(b,1),(b,2)$$

B
$$(1,a),(2,b),(b,1),(b,2)$$

C
$$(a,1),(a,2),(1,b),(2,b)$$

D
None of the above

Solution

Given : $$S=\{ 1,2\} ,T=\{ a,b\} $$
$$T\times S$$ is the set of ordered pair of elements of $$T$$ and $$S$$ respectively
Then, $$T\times S=\{a,b\}\times \{1,2\}$$
$$T\times S=\{ (a,1),(a,2),(b,1),(b,2)\} $$
Question 3
1 / -0

n(A)=m and n(B)=n ; then

A
n(A)+n(B)=n(A+B)

B
n(A)-n(B)=n(A+B)

C
A$$\times$$B=mn

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

Solution

C is not even logical while in the first 2 cases:

If there are elements common in A and B L.H.S of the first statement will be greater than R.H.S.

Clearly option B is also untrue as the combination of A and B cannot have lesser number of elements than A alone.

The last option has to be true as cartesian product of 2 sets gives a matrix having n(A) and n(B) as columns and rows.whose product will give the number of elements in the matrix.
Question 4
1 / -0

$$\left (A \cap B \right ) \times C$$

A
$$\left (A \times B \right ) \cap \left (B \times C \right )$$

B
$$\left (A \times C \right ) \cap \left (B \times C \right )$$

C
$$\left (A \times B \right ) \cup \left (B \times C \right )$$

D
$$\left (A \times B \right ) \cup \left (A \times C \right )$$

Solution
Question 5
1 / -0

Given $$M=\{5,6,7\}$$ and $$N=\{6,8,10\}$$ find element of $$(M\cup N)\times N$$

A
$$\{5,6\}$$

B
$$\{5,8\}$$

C
$$\{5,10\}$$

D
All of the above

Solution

For two non-empty sets $$A$$ and $$B$$, the Cartesian product $$A\times B$$ is the set of all ordered pair of elements from $$A$$ and $$B$$.
$$A \times B = \{(x, y) : x \in A, y \in B\}$$

If $$A= \{a,b\}$$ and $$B= \{x, y\},$$
then $$A\times B = \{(a, x); (a, y); (b, x); (b, y)\}$$
here, $$A=M\cup N$$ and $$B=N$$
$$A=M\cup N$$ ={ $$5,6,7,8,10$$ }
$$B=$${$$6,8,10$$}
$$A\times B = (M\cup N)\times N=(A,6),(A,8),(A,10)$$
hence A,B,C all are correct so answer is D
Question 6
1 / -0

Given $$A=\{b,c,d\}$$ and $$B=\{x,y\}$$ : find element of $$A\times B$$ .

A
$$\{b,x\}$$

B
$$\{b,y\}$$

C
$$\{c,x\}$$

D
All of the above

Solution

Given $$A=\{b,c,d\}$$ and $$B=\{x,y\}$$, then
$$A\times B=\{(b,x),(c,x),(d,x),(b,y),(c,y),(d,y)\}$$
Hence, all the elements given in options are elements of $$A\times B$$.
Question 7
1 / -0

$$n(A)=4 $$ and $$n(B) =5$$: $$n(A \times B)=$$

A
$$20$$

B
$$10$$

C
$$30$$

D
None of the above

Solution

If $$n(A)=m$$,and $$n(B)=n$$,then $$n(A\times B)=mn$$
so$$n(A\times B)=5.4=20$$
Question 8
1 / -0

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

A
n(A) x n(B)

B
n(A $$\bigcap$$ B)

C
n(A $$\bigcup$$ B)

D
all of these

Solution
Question 9
1 / -0

Which one of the statement is false ?

A
$$\phi \times A = \phi$$

B
A $$\times$$ B = B $$\times$$ A

C
A $$\times$$ B = {(x $$\times$$ y) : x A and y B}

D
$$R^{-1}$$ = {(y, x) : (x, y) R}
Question 10
1 / -0

R is a relation in A and (a, b) $$\notin$$ r, implies (b, a) $$\notin$$ R then R is said to be ____ relation

A
symmetric

B
contradictory

C
skew symmetric

D
none of these

Solution

$$R$$ is a relation in $$A$$ and $$(a,b)$$.
$$\Rightarrow (b,a)\notin R$$
Hence $$'R'$$ is a skew symmetric relation.
Answer:-$$C$$

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Re-Attempt Weekly Quiz Competition

Relations Test 11

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Relations Test 11

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SHARING IS CARING

Question 1

$$\{b,b\}$$

$$\{b,c\}$$

$$\{b,d\}$$

All of the above

Solution

Question 2

$$(a,1),(a,2),(b,1),(b,2)$$B×A={(a,1),(a,2),(b,1),(b,2)}

$$(1,a),(2,b),(b,1),(b,2)$$

$$(a,1),(a,2),(1,b),(2,b)$$

None of the above

Solution

Question 3

n(A)+n(B)=n(A+B)

n(A)-n(B)=n(A+B)

A$$\times$$B=mn

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

Solution

Question 4

$$\left (A \times B \right ) \cap \left (B \times C \right )$$

$$\left (A \times C \right ) \cap \left (B \times C \right )$$

$$\left (A \times B \right ) \cup \left (B \times C \right )$$

$$\left (A \times B \right ) \cup \left (A \times C \right )$$

Solution

Question 5

$$\{5,6\}$$

$$\{5,8\}$$

$$\{5,10\}$$

All of the above

Solution

Question 6

$$\{b,x\}$$

$$\{b,y\}$$

$$\{c,x\}$$

All of the above

Solution

Question 7

$$20$$

$$10$$

$$30$$

None of the above

Solution

Question 8

n(A) x n(B)

n(A $$\bigcap$$ B)

n(A $$\bigcup$$ B)

all of these

Solution

Question 9

$$\phi \times A = \phi$$

A $$\times$$ B = B $$\times$$ A

A $$\times$$ B = {(x $$\times$$ y) : x A and y B}

$$R^{-1}$$ = {(y, x) : (x, y) R}

Question 10

symmetric

contradictory

skew symmetric

none of these

Solution

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$$(a,1),(a,2),(b,1),(b,2)$$