University Maths Solution

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Maths Question
Question 1

$\begin{align}  & \text{In the sequel }\mathbb{N},\mathbb{Z}\text{ and }\mathbb{Q}\text{ denote the set of all} \\ &\text{natural numbers, the set of all integers and the set of all rational numbers} \\ & \text{Exhibit in tabular form} \\ & (a)A=\{x:x\in \mathbb{N},\,{{x}^{2}}-5x+6=0\} \\ & (b)B=\{y:y\in \mathbb{Z},2{{y}^{2}}-3y+1=0\} \\ & (c)C=\{x:x\in \mathbb{Q},6{{x}^{2}}-13x+6=0\} \\ & (d)D=\{x:x\in \mathbb{Q},{{x}^{2}}-x+1=0\} \\ & (e)\text{the set of all integers whose squares are less than 20} \\ & \text{(f) the set of all integers between }-3\text{ and }+3 \\ & g)P=\{x\in \mathbb{N}:-1\le x\le 1\} \\\end{align}$

Question 2

$\begin{align}  & \text{Let }A=\{a,b,c,d,e\},\text{ }B=\{a,d,e,f,g\}\text{ and } \\ & C=\{a,c,ef,g\} \\ & \text{compute} \\ & (a)\text{ }A\cap B \\ & (b)A\cap C \\ & (c)B\cap C \\ & (d)A\cup B \\ & (e)A\cup C \\ & (f)B\cup C \\ & (g)(A\cup B)\cup C \\ & (h)(A\cap B)\cap C \\ & (i)A\cap (B\cap C) \\ & (j)A\cup (B\cap C) \\\end{align}$

Question 3

$\begin{align}  & \text{Let }\mu =\{1,2,3,\cdot \cdot \cdot ,10\},\text{ }X=\{2,4,6,8\},Y=\{1,2,3,4\} \\ & \text{and }Z=\{3,4,5,6,8\} \\ & (a)\text{ Exhibit in tabular form the set }X',Y'\text{ and }Z' \\ & (b)\text{ Verify the identities } \\ & (X\cup Y)'=X'\cap Y' \\ & (Y\cap Z)'=Y'\cup Z' \\\end{align}$

Question 4

$\begin{align}  & \text{Let }P,Q\text{ and }R\text{ be set, prove that } \\ & \text{(i)  }(P\cup Q)\cup R=P\cup (Q\cup R) \\ & (ii)\text{ }(P\cap Q)\cap R=P\cap (Q\cap R) \\\end{align}$

Question 5

$\begin{align}  & \text{Let }P,Q\text{ and }R\text{ be set, prove that } \\ & (i)P\cap (Q\cup R)=(P\cap Q)\cup (P\cap R) \\ & (i)P\cup (Q\cap R)=(P\cup Q)\cap (P\cup R) \\\end{align}$

Question 6

$\text{Prove rigourosuly }(X\cup Y)'=X'\cap Y'$

Question 7

$\text{Prove rigourosuly }(X\cap Y)'=X'\cup Y'$

Question 8

$\text{Prove that }Z-(X\cup Y)=(Z-X)\cap (Z-Y)$

Question 9

$\text{Prove that }X\cap (Y-Z)=(X\cap Y)-(X\cap Z)$

Question 10

$\text{Prove that (}X-Y)\cup (X-Z)=X-(Y\cap Z)$

Question 11

$Prove\text{ }that\text{ }(X-Y)\cap (X-Z)=X-(Y\cup Z)$

Question 12

$\text{Prove that }(X-Y)-Z=X-(Y\cup Z)$

Question 13

$\text{Prove that }(X-Y)-Z=(X-Y)\cap (X-Z)$

Question 14

$\text{Prove that }(X-Y)\cup (Y-X)=(X\cup Y)-(X\cap Y)$

Question 15

$\text{prove that }(X-Y)\cup (Y-X)=(X\cup Y)\cap (X'\cup Y')$

Question 16

$\begin{align}  & \text{Prove that } \\ & (i)\text{ (}X\cap Y)=X\Leftrightarrow X\subseteq Y \\ & (ii)X\cup Y=X\Leftrightarrow X\supseteq Y \\\end{align}$ 

Question 17

$\begin{align}  & \text{Simplify } \\ & \text{i) }A\cup (A'\cap B) \\ & ii)A'\cap (A\cup B') \\ & iii)(A'\cap B')\cup (A\cap B) \\\end{align}$

Question 18

$\begin{align}  & a)\text{ If }n(A)\,\text{denotes the number of elements contained in the set }A,\text{ prove that } \\ & n(A)+n(B)=n(A\cup B)+n(A\cap B) \\ & (b)\text{Prove that } \\ & n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C) \\\end{align}$

Question 19

Suppose the following data represents a survey of 500 students who were enrolled in a freshman college mathematics course

450 passed the course

10 of those who failed the course liked it,

25 of those who failed the course signed up for another mathematics course,

55 of those who liked the course signed up for another mathematics course

60 of those who passed the course signed up for another mathematics course

350 of those who passed the course liked it,

300 of those who passed the course liked it but did not sign up for another mathematics course

 

(a) Give a Venn diagram summarizing this data

(b)  How many of the students who failed disliked the course and did not sign up for another mathematics course

(c) How many of the students liked the course?

(d) How many of the students who did not like the course passed it?