Tag: mathematics and statistics

Quizzes (38) Questions (955)

Questions Related to mathematics and statistics

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

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

  1. $\left \{(8,3), (8,2), (8,1), (8,8)\right \}$
  2. $\left \{(1,2), (2,2), (3,3), (8,8)\right \}$
  3. $\left \{(3,1), (3,2), (3,3), (3,8)\right \}$
  4. $\left \{(1,3), (2,3), (3,3), (8,3)\right \}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

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

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

What is the Cartesian product of $A = \left {1, 2\right }$ and $B = \left {a, b\right }$?

  1. $\left \{(1, a), (1, b), (2, a), (b, b)\right \}$
  2. $\left \{(1, 1), (2, 2), (a, a), (b, b)\right \}$
  3. $\left \{(1, a), (2, a), (1, b), (2, b)\right \}$
  4. $\left \{(1, 1), (a, a), (2, a), (1, b)\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If $A $ and $B$ are two non empty sets, then the Cartesian product $A \times B$ is set of all ordered pairs $(a,b)$ such that $a\in A$ and $b\in B$.


Given $A ={1,2}$ and $B = {a,b}$

Hence $A\times B = {(1,a),(1,b),(2,a),(2,b)}$ 

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

Let a relation $R$ be defined by $R=\left {(4,5), (1,4), (4,6), (7,6), (3,7)\right }$. The relation $R^{-1}\circ R$ is given by

  1. $\left \{(1,1), (4,4), (7,4), (4,7), (7,7)\right \}$
  2. $\left \{(1,1), (4,4), (4,7), (7,4), (7,7),(3,3)\right \}$
  3. $\left \{(1,5), (1,6), (3,6)\right \}$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
We have $R=\left \{(4,5), (1,4), (4,6), (7,6), (3,7)\right \}$.
$\therefore R^{-1}=\left \{(5,4), (4,1), (6,4), (6,7), (7,3)\right \}$
$(4,4)\in R^{-1}\circ R$ because $(4,5)\in R$ and $(5,4)\in R^{-1}$
$(1,1)\in R^{-1}\circ R$ because $(1,4)\in R$ and $(4,1)\in R^{-1}$
$(4,4)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,4)\in R^{-1}$
$(4,7)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,7)\in R^{-1}$
$(7,4)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,4)\in R^{-1}$
$(7,7)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,7)\in R^{-1}$
$(3,3)\in R^{-1}\circ R$ because $(3,7)\in R$ and $(7,3)\in R^{-1}$
$\therefore R^{-1}\circ R=\left \{(4,4), (1,1), (4,7), (7,4), (7,7), (3,3)\right \}$.
$\therefore$ The correct answer is $B$.
Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Given $(a - 2, b + 3) = (6, 8)$, are equal ordered pair. Find the value of $a$ and $b$.

  1. $a = 8$ and $b = 5$
  2. $a = 8$ and $b = 3$
  3. $a = 5$ and $b = 5$
  4. $a = 8$ and $b = 6$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By equality of ordered pairs, we have
$(a - 2, b + 3) = (6, 8)$
On equating we get
$a - 2 = 6$
$a = 8$
$b + 3 = 8$
$b = 5$
So, the value of$ a = 8 $ and $ b = 5.$