Tag: basic operations on sets

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Questions Related to basic operations on sets

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If x belongs to set of integers, A is the solution set of $2(x-1)< 3x-1$ and B is the solution set of $4x-3\leq 8+x$, find A$\cap$B.

  1. $\left\{0, 1, 2\right\}$
  2. $\left\{1, 2, 3\right\}$
  3. $\left\{0, 1, 2, 3\right\}$
  4. $\left\{0, 2, 4\right\}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$2(x-1)< 3x-1$
$\Rightarrow 2x-2< 3x-1$
$\Rightarrow 2x-3x< -1+2$
$\Rightarrow -x< 1$
$\Rightarrow x> -1$

$4x-3\leq 8+x$
$4x-3-x+3\leq 8+x-x+3$
$\Rightarrow 4x-x\leq 8+3$
$\Rightarrow 3x\leq 11$
$x\leq \dfrac{11}{3}\Rightarrow x\le 3.67$
Then $-1< x\leq 3.67$
Then, $A\cap B$ is $ \{0,1,2,3 \}$
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A = \left {2, 3, 4, 8, 10\right }, B = \left (3, 4, 5, 10, 12\right }, C = \left {4, 5, 6, 12, 14\right }$, then $(A\cap B)\cup (A\cap C)$ is equal to

  1. $\left \{3, 4, 10\right \}$
  2. $\left \{2, 9, 10\right \}$
  3. $\left \{4, 5, 6\right \}$
  4. $\left \{3, 5, 14\right \}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ A={2,3,4,8,10} , B={3,4,5,10,12} , C={4,5,6,12,14}$


To find : $(A\cap B)$ $\cup  (A\cap C)$

$A\cap B$ = elements common in sets $A$ and $B$. 
$A\cap C$ = elements common in sets $A$ and $C$ 

$A\cap C = \{4\}$ 
$(A\cap B)\cup (A\cap C)=$elements in $(A\cap B)$ and $(A\cap C)$
$(A\cap B) \cup  (A\cap C) = \{3,4,10\} \cup \{4\} $
$(A\cap B) \cup  (A\cap C)={3,4,10}$

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Which is the simplified representation of 
$\left( {{A^/} \cap \,{B^/} \cap \,C} \right) \cup \left( {B\, \cap \,C} \right) \cup \left( {A \cap \,C} \right)$  

where A,B,C are subsets of X

  1. A

  2. B

  3. C
  4. $X \cap \,\left( {A \cup B \cup C} \right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using set theory laws, simplify the expression ((A' intersect B') intersect C) union (B intersect C) union (A intersect C). Notice that C is a common factor across the union terms: (A' intersect B') intersect C can be written as (A union B)' intersect C. Distributing C across the union with other terms using set identities yields simply C.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $aN=\left{ ax:x\epsilon N \right}$, then the set $3N\cap 7N$ is

  1. $21\ N$
  2. $10\ N$
  3. $4\ N$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set nN represents multiples of n. The intersection of multiples of 3 (3N) and multiples of 7 (7N) consists of multiples of the least common multiple of 3 and 7, which is 21. Thus, 3N intersect 7N = 21N.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

A is a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen. the set $A$ is reconstructed by replacing the elements of $P.A$ subset $Q$ of $A$ is again chosen. the number of ways of choosing $P$ and $Q$ so that $P \cap Q$

  1. $9. ^{n}C _{2}$
  2. $3^{n}- ^{n}C _{2}$
  3. $^{n}C _{2}.3^{n-2}$
  4. $4^{n}-3^{n}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $A=\left{ a,b,c,d \right} ,B=\left{ b,c,d,e \right}$. Then $n\left[ \left( A\times B \right) \cap \left( B\times A \right)  \right]$ is equal to 

  1. $3$
  2. $6$
  3. $9$
  4. $none$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The number of elements in the Cartesian product of two sets is the product of their cardinalities. Since both sets A and B have 4 elements, the intersection of A cross B and B cross A consists of the ordered pairs (x, y) where both x and y belong to the intersection of A and B. The intersection of A and B contains 3 elements, so the number of such pairs is 3 squared, which is 9.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

The set of all points where the function $f(x)=||x|$ is twice differentiable is

  1. $(-\infty, \infty)$
  2. $(-\infty, 0)\cup (0, \infty)$
  3. $(0, \infty)$
  4. $[0, \infty)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The function can be simplified based on the absolute value definition. For x different from zero, the function equals x squared or negative x squared, which is infinitely differentiable. At x equals zero, the first derivative exists but the second derivative does not exist, so the function is twice differentiable everywhere except at zero.