Sec 4 E Maths: Sets and Venn Diagrams, practice questions & worked solutions

Given that $\xi=\{x:x\text{ is an integer},\ 1\le x\le12\}$, and

$$A=\{x:x\text{ is a factor of }6\}$$

$$B=\{x:\sqrt{x}<3\}$$

Find the value of $n(A'\cap B)'$. [2]

1. Use set notation to describe the shaded region.

   

   [1]

2. $$\begin{aligned}\xi&=\{x:x\text{ is an integer and }1\leq x\leq10\}\\A&=\{x:x\text{ is a factor of }10\}\\B&=\{x:x\text{ is a prime number}\}\end{aligned}$$

   1. List the elements contained in the set $A\cap B$. [1]

   2. Find the value of $\mathrm n(A\cup B')$. [1]

1. On the Venn diagram, shade the region which represents $A'\cup B$.

   

   [1]

2. $\varepsilon=\{\text{integers }x:1\le x<10\}$

   $P=\{\text{multiples of }3\}$

   $Q=\{\text{perfect squares}\}$

   1. List the elements in $P'\cap Q'$. [1]

   2. A number, $y$ is chosen at random from the set $P\cup Q$.

      Find the probability that $y\notin P$. [1]

1. On the Venn diagram shown in the answer space, shade the set $A\cap B'$. [1]

   

2. $\xi=\{x:x\text{ is an integer, }1\le x\le p\text{ and }p<50\}$
   $S=\{x:x\text{ is a multiple of 2}\}$
   $T=\{x:x\text{ is a perfect square}\}$

   1. List the elements of $S\cap T$ such that there are only 3 elements in it. [1]

   2. Find the largest $p$ such that $S\cap T=\varnothing$. [1]

Given that $\xi=\{\text{integers }x:\ p\le x<20\}$
$A=\{\text{factors of }24\}$
$B=\{\text{prime numbers}\}$
$C=\{\text{perfect squares}\}$

1. List the elements of $A$ if $p=2$. [1]

2. For $p>10$, list the elements of $(B\cup C)'$ such that $\mathrm n(B\cup C)'=4$. [1]

3. Find the smallest $p$ such that $A\cap C=\varnothing$. [1]

$$\mathcal E=\{x:x\text{ is an integer and }1\leq x<25\}$$

$$P=\{x:x\text{ is a multiple of }3\}$$

$$R=\{x:(x-3)(x-7)=0\}$$

1. List the elements contained in the set $P\cap R'$. [2]

2. A number $q$ is chosen at random from the universal set.
   Find the probability that $q\in P\cup R$. [2]

1. $\xi=\left\{\text{integers }x:2\le x\le18\right\}$
   $M=\left\{\text{prime numbers}\right\}$
   $N=\left\{\text{factors of }28\right\}$

   Draw a Venn diagram to illustrate this information. [2]

2. Use set notation to describe the shaded region.

   

   [1]

3. The sets $\xi$, $P$ and $Q$ satisfy the conditions $n(\xi)=18$, $n(P)=9$ and $n(Q)=15$.

   Find the largest possible value of $n(P\cup Q)'$. [1]

$\xi=\{\text{integers }x:1\le x\le16\}$

$V=\{1,2,4,8,16\}$

$W=\{1,4,7,9,10,14\}$

Some of the information is shown on the Venn diagram.

1. Complete the Venn diagram by representing all the elements in the given sets.

   Answer in above Venn diagram [1]

2. Describe the elements of Set $V$. [1]

3. List the elements contained in the set $V'\cap W$ . [1]

4. Find the number of elements in $(V\cap W')\cup(V\cup W)'$ . [1]

5. Use one of the symbols below to complete each statement.

   $$\quad=\qquad\varnothing\qquad\subset\qquad\not\subset\qquad\notin\qquad\in\qquad\xi$$

   1. $\{2,16\}$  ………  $V$ [1]

   2. $13$  ………  $W$ [1]

1. Solve the inequality $-11\leq4-3x\leq10$. [2]

2. $\xi=\{\text{integers }x:-3\leq x\leq7\}$
   $A=\{x:-11\leq4-3x\leq10\}$
   $B=\{\text{prime numbers}\}$

   1. Complete the Venn diagram to illustrate this information.

      

      [2]

   2. Find the number of elements in the set $(A\cap B)'$. [1]

   3. List the elements in the set $A'\cap B'$. [1]

   4. Use one of the symbols below to complete each statement.

      $$\varnothing\quad\subset\quad\not\subset\quad\in\quad\notin$$

      1. $4\qquad\rule{1.4cm}{.4pt}\qquad B$ [1]

      2. $\{2,3\}\qquad\rule{1.4cm}{.4pt}\qquad B$ [1]

Given that
$\varepsilon=\{\text{positive integers less than }17\}$,
$X=\{2,3,4\}$
$Y=\{\text{prime numbers}\}$
$Z=\{2,3,4,5,\ldots,n-1,n\}$

1. Draw a Venn Diagram to represent the set $\varepsilon$, $X$ and $Y$, indicating the elements clearly. [3]

2. 1. List all subsets of $X$. [2]

   2. Hence, deduce the number of subsets of $Z$ in term of $n$. [2]