Tag: cartesian product of sets

Questions Related to cartesian product of sets

$M={0,1,2}$ and $N={1,2,3}$: find (N-M) $\times$(N $\cap$M)

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. ${3,1}$

  2. ${3,2}$

  3. ${3,3}$

  4. None of the above

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Correct Option: A,B
Explanation:

For two sets, A and B the set difference of set B from set A is the set of all element in A but not in B.
Example:
$A={a,b,c,d}$
$B={c,d,e}$
$A-B={a,b}$

$M={0,1,2}$
$N={1,2,3}$
$N-M={3}$
$N \cap M={1,2}$
$(N-M)\times (N\cap M)$ $={3,1},{3,2}$

Given M={0,1,2} and N={1,2,3}, then (M $\cup$ N) $\times$(M-N) contains

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. ${0,0}$

  2. ${1,0}$

  3. ${2,0}$

  4. ${3,0}$

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Correct Option: A,B,C,D
Explanation:
$M=\{0,1,2\}$
$N=\{1,2,3\}$
$M\cup N=\{0,1,2,3\}$
$M-N=\{0\}$
$(M\cup N)\times (M-N)$ $=\{0,0\},\{1,0\},\{2,0\},\{3,0\}$

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

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. ${b,b}$

  2. ${b,c}$

  3. ${b,d}$

  4. All of the above

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Correct Option: D
Explanation:

$A\times A \Rightarrow$ 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 $A \times A$

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

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $20$

  2. $10$

  3. $30$

  4. None of the above

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Correct Option: A
Explanation:

If $n(A)=m$,and $n(B)=n$,then $n(A\times B)=mn$

so$n(A\times B)=5.4=20$

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

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. n(A)+n(B)=n(A+B)

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

  3. A$\times$B=mn

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

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Correct Option: D
Explanation:

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.

n (A $\times$ B) =

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. n(A) x n(B)

  2. n(A $\bigcap$ B)

  3. n(A $\bigcup$ B)

  4. all of these

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Correct Option: A

$\left (A \cap B  \right ) \times C$

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

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

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

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

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

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Correct Option: B

$\left (A \cap B  \right ) \times C$


mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

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

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

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

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

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Correct Option: A

Which one of the statement is false ?

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $\phi \times A = \phi$

  2. A $\times$ B = B $\times$ A

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

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

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Correct Option: B

A $\times$ (B - C) =

mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

  1. $(A \times B) - (A \times C)$

  2. $(A \times C) - (A \times B)$

  3. $(A \times B) \bigcup (A \times C)$

  4. $(A \times B) \bigcap (A \times C)$

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Correct Option: A