9th Class Math Chapter 3 Notes

Q1(i) — Set-builder: $\{1,4,9,16,25,36,\dots ,484\}$

Write in set-builder notation.

1. Each term is a perfect square: $1^2,2^2,\dots ,22^2$ (since $22^2=484$).

Answer: $\{x\mid x=n^2,n\in N\wedge 1\le n\le 22\}$

Q1(ii) — Set-builder: $\{2,4,8,16,\dots ,256\}$

Write in set-builder notation.

1. Powers of $2$: $2^1,\dots ,2^8$ (since $2^8=256$).

Answer: $\{x\mid x=2^n,n\in N\wedge 1\le n\le 8\}$

Q1(iii) — Set-builder: $\{0,\pm 1,\pm 2,\dots ,\pm 1000\}$

Write in set-builder notation.

1. All integers from $-1000$ to $1000$.

Answer: $\{x\mid x\in Z\wedge -1000\le x\le 1000\}$

Q1(iv) — Set-builder: $\{6,12,18,\dots ,120\}$

Write in set-builder notation.

1. Multiples of $6$: $6(1),6(2),\dots ,6(20)$.

Answer: $\{x\mid x=6n,n\in N\wedge 1\le n\le 20\}$

Q1(v) — Set-builder: $\{100,102,\dots ,400\}$

Write in set-builder notation.

1. Even numbers from $100$ to $400$.

Answer: $\{x\mid x\text{ is even}\wedge 100\le x\le 400\}$

Q1(vi) — Set-builder: $\{1,3,9,27,81,\dots \}$

Write in set-builder notation.

1. Powers of $3$ starting from $3^0$, continuing indefinitely.

Answer: $\{x\mid x=3^n,n\in W\}$

Q1(vii) — Set-builder: $\{1,2,4,5,10,20,25,50,100\}$

Write in set-builder notation.

1. Every listed number divides $100$ exactly.

Answer: $\{x\mid x\text{ is a divisor of }100\}$

Q1(viii) — Set-builder: $\{5,10,15,\dots ,100\}$

Write in set-builder notation.

1. Multiples of $5$: $5(1),\dots ,5(20)$.

Answer: $\{x\mid x=5n,n\in N\wedge 1\le n\le 20\}$

Q1(ix) — Set-builder: integers between $-100$ and $1000$

Write in set-builder notation.

1. "Between" is taken as strict (exclusive) bounds.

Answer: $\{x\mid x\in Z\wedge -100<x<1000\}$

Q2(i) — Tabular: $\{x\mid x\text{ mult. of }3\wedge x\le 36\}$

Write in tabular form.

1. Multiples of $3$ from $3$ up to $36$.

Answer: $\{3,6,9,12,15,18,21,24,27,30,33,36\}$

Q2(ii) — Tabular: $\{x\mid x\in R\wedge 2x+1=0\}$

Write in tabular form.

1. Solve $2x+1=0\Rightarrow x=-\frac{1}{2}$; this is a real number, so it belongs.

Answer: $\left\{-\frac{1}{2}\right\}$

Q2(iii) — Tabular: $\{x\mid x\in P\wedge x<12\}$

Write in tabular form ($P$ = prime numbers).

1. Primes less than $12$: $2,3,5,7,11$.

Answer: $\{2,3,5,7,11\}$

Q2(iv) — Tabular: $\{x\mid x\text{ divides }128\}$

Write in tabular form.

1. $128=2^7$; its divisors are the powers of $2$ up to itself.

Answer: $\{1,2,4,8,16,32,64,128\}$

Q2(v) — Tabular: $\{x\mid x=2^n,n\in N\wedge n<8\}$

Write in tabular form.

1. $n=1,\dots ,7$.

Answer: $\{2,4,8,16,32,64,128\}$

Q2(vi) — Tabular: $\{x\mid x\in N\wedge x+4=0\}$

Write in tabular form.

1. $x=-4$, which is not a natural number.

Answer: $\{\}=\varnothing$

Q2(vii) — Tabular: $\{x\mid x\in N\wedge x=x\}$

Write in tabular form.

1. The condition $x=x$ is true for every natural number.

Answer: $\{1,2,3,4,\dots \}=N$

Q2(viii) — Tabular: $\{x\mid x\in Z\wedge 3x+1=0\}$

Write in tabular form.

1. $x=-\frac{1}{3}$, which is not an integer.

Answer: $\{\}=\varnothing$

Q3(i) — Two proper subsets of $\{a,b,c\}$

Give two proper subsets.

1. A proper subset contains some but not all elements.

Answer: $\{a\}$ and $\{b,c\}$ (for example)

Q3(ii) — Two proper subsets of $\{0,1\}$

Give two proper subsets.

1. Any subset other than $\{0,1\}$ itself.

Answer: $\{0\}$ and $\{\}$

Q3(iii) — Two proper subsets of $N$

Give two proper subsets.

1. Any set strictly contained in $N$.

Answer: $\{1,2\}$ and the set of even naturals $\{2,4,6,\dots \}$

Q3(iv) — Two proper subsets of $Z$

Give two proper subsets.

1. Any set strictly contained in $Z$.

Answer: $N$ and $\{0\}$

Q3(v) — Two proper subsets of $Q$

Give two proper subsets.

1. Any set strictly contained in $Q$.

Answer: $Z$ and $N$

Q3(vi) — Two proper subsets of $R$

Give two proper subsets.

1. Any set strictly contained in $R$.

Answer: $Q$ and $Z$

Q3(vii) — Two proper subsets of $\{x\mid x\in Q\wedge 0<x\le 2\}$

Give two proper subsets.

1. Any set strictly contained in the given set.

Answer: $\{1\}$ and $\{x\mid x\in Q\wedge 0<x\le 1\}$

Q4(i) — Number of elements in $P(\{\})$

Find the number of elements of the power set.

1. $n(S)=0\Rightarrow n(P(S))=2^0$.

Answer: $1$

Q4(ii) — Number of elements in $P(\{0,1\})$

Find the number of elements of the power set.

1. $n(S)=2\Rightarrow 2^2$.

Answer: $4$

Q4(iii) — Number of elements in $P(\{1,\dots ,7\})$

Find the number of elements of the power set.

1. $n(S)=7\Rightarrow 2^7$.

Answer: $128$

Q4(iv) — Number of elements in $P(\{0,1,\dots ,7\})$

Find the number of elements of the power set.

1. $n(S)=8\Rightarrow 2^8$.

Answer: $256$

Q4(v) — Number of elements in $P(\{a,\{b,c\}\})$

Find the number of elements of the power set.

1. This set has $2$ elements ($a$ and $\{b,c\}$, the latter counted once) $\Rightarrow 2^2$.

Answer: $4$

Q4(vi) — Number of elements in $P(\{\{a,b\},\{b,c\},\{d,e\}\})$

Find the number of elements of the power set.

1. This set has $3$ elements (each itself a set) $\Rightarrow 2^3$.

Answer: $8$

Q5(i) — Power set of $\{9,11\}$

Write down the power set.

1. List every subset, including $\varnothing$ and the set itself.

Answer: $P(S)=\{\varnothing ,\{9\},\{11\},\{9,11\}\}$

Q5(ii) — Power set of $\{+,-,\times ,÷\}$

Write down the power set.

1. $4$ elements give $2^4=16$ subsets.

Answer: $P(S)=\{\varnothing ,\{+\},\{-\},\{\times \},\{÷\},\{+,-\},\{+,\times \},\{+,÷\},\{-,\times \},\{-,÷\},\{\times ,÷\},\{+,-,\times \},\{+,-,÷\},\{+,\times ,÷\},\{-,\times ,÷\},\{+,-,\times ,÷\}\}$

Q5(iii) — Power set of $\{\varnothing \}$

Write down the power set.

1. This set has one element, namely $\varnothing$ itself.

Answer: $P(S)=\{\varnothing ,\{\varnothing \}\}$

Q5(iv) — Power set of $\{a,\{b,c\}\}$

Write down the power set.

1. Two elements: $a$ and $\{b,c\}$.

Answer: $P(S)=\{\varnothing ,\{a\},\{\{b,c\}\},\{a,\{b,c\}\}\}$

Q6(i) — List $A$ and $B$ (multiples of 6 and 8, within $U$)

$U=\{x\mid x\text{ mult. of }2\wedge 0<x\le 30\}$, $A=\{x\mid x\text{ mult. of }6\}$, $B=\{x\mid x\text{ mult. of }8\}$ (both within $U$). List $A$ and $B$.

1. Multiples of $6$ up to $30$: $6,12,18,24,30$.
2. Multiples of $8$ up to $30$: $8,16,24$.

Answer: $A=\{6,12,18,24,30\}$, $B=\{8,16,24\}$

Q6(ii) — Find $A\cap B$

Using the sets above, find $A\cap B$.

1. Common element(s) between $\{6,12,18,24,30\}$ and $\{8,16,24\}$.

Answer: $A\cap B=\{24\}$

Q6(iii) — Venn diagram for $U,A,B$

Draw a Venn diagram for the sets above.

1. Rectangle $U=\{2,4,\dots ,30\}$; circle $A$ holds $6,12,18,30$ (not shared) plus $24$ in the overlap; circle $B$ holds $8,16$ (not shared) plus $24$ in the overlap; outside both circles but inside $U$: $2,4,10,14,20,22,26,28$.

Answer: Overlap $=\{24\}$; $A$-only $=\{6,12,18,30\}$; $B$-only $=\{8,16\}$

Q7(i) — List $G$ and $H$

$U=\{x\mid x\in Z\wedge 0<x\le 150\}$, $G=\{x\mid x=2^m\}$, $H=\{x\mid x\text{ is a square}\}$ (both within $U$). List $G$ and $H$.

1. Powers of $2$ up to $150$: $2,4,8,16,32,64,128$.
2. Perfect squares up to $150$: $1,4,9,16,25,36,49,64,81,100,121,144$.

Answer: $G=\{2,4,8,16,32,64,128\}$, $H=\{1,4,9,16,25,36,49,64,81,100,121,144\}$

Q7(ii) — Find $G\cup H$

Find $G\cup H$.

1. Combine both lists, removing duplicates ($4,16,64$ appear in both).

Answer: $\{1,2,4,8,9,16,25,32,36,49,64,81,100,121,128,144\}$

Q7(iii) — Find $G\cap H$

Find $G\cap H$.

1. Elements common to both lists: $4,16,64$.

Answer: $G\cap H=\{4,16,64\}$

Q8(i) — Find $P\cap Q$

$P=\{x\mid x\text{ prime}\wedge 0<x\le 20\}$, $Q=\{x\mid x\text{ divides }210\wedge 0<x\le 20\}$. Find $P\cap Q$.

1. $P=\{2,3,5,7,11,13,17,19\}$.
2. $210=2\times 3\times 5\times 7$; divisors $\le 20$: $Q=\{1,2,3,5,6,7,10,14,15\}$.
3. Common elements: $2,3,5,7$.

Answer: $P\cap Q=\{2,3,5,7\}$

Q8(ii) — Find $P\cup Q$

Find $P\cup Q$.

1. Combine $P=\{2,3,5,7,11,13,17,19\}$ and $Q=\{1,2,3,5,6,7,10,14,15\}$, removing duplicates.

Answer: $\{1,2,3,5,6,7,10,11,13,14,15,17,19\}$

Q9(i) — Verify commutativity for $A=\{1,2,3,4,5\}$, $B=\{4,6,8,10\}$

Verify $A\cup B=B\cup A$ and $A\cap B=B\cap A$.

1. $A\cup B=\{1,2,3,4,5,6,8,10\}$ and $B\cup A$ gives the same set — order doesn't matter.
2. $A\cap B=\{4\}$ and $B\cap A=\{4\}$ — also the same.

Answer: Both properties hold: $A\cup B=B\cup A=\{1,2,3,4,5,6,8,10\}$; $A\cap B=B\cap A=\{4\}$

Q9(ii) — Verify commutativity for $N,Z$

Verify $N\cup Z=Z\cup N$ and $N\cap Z=Z\cap N$.

1. Since $N\subset Z$: $N\cup Z=Z$ (same either order); $N\cap Z=N$ (same either order).

Answer: Both properties hold: union $=Z$, intersection $=N$

Q9(iii) — Verify commutativity for $A=\{x\mid x\in R\wedge x\ge 0\}$, $B=R$

Verify $A\cup B=B\cup A$ and $A\cap B=B\cap A$.

1. Since $A\subset R$: $A\cup R=R$ (either order); $A\cap R=A$ (either order).

Answer: Both properties hold: union $=R$, intersection $=A$

Q10 — Verify De Morgan's Laws

$U=\{a,\dots ,j\}$, $A=\{a,b,c,d,g,h\}$, $B=\{c,d,e,f,j\}$. Verify De Morgan's laws and draw a Venn diagram.

1. $A^{\prime }=U-A=\{e,f,i,j\}$, $B^{\prime }=U-B=\{a,b,g,h,i\}$.
2. $A\cup B=\{a,b,c,d,e,f,g,h,j\}$, so $(A\cup B{)}^{\prime }=\{i\}$; and $A^{\prime }\cap B^{\prime }=\{i\}$ — matches.
3. $A\cap B=\{c,d\}$, so $(A\cap B{)}^{\prime }=\{a,b,e,f,g,h,i,j\}$; and $A^{\prime }\cup B^{\prime }=\{a,b,e,f,g,h,i,j\}$ — matches.
4. Venn diagram: rectangle $U$; circles $A$ and $B$ overlapping at $\{c,d\}$; $A$-only $=\{a,b,g,h\}$; $B$-only $=\{e,f,j\}$; outside both $=\{i\}$.

Answer: $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }=\{i\}$; $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }=\{a,b,e,f,g,h,i,j\}$ — both laws verified

Q11(i) — Verify $A\cup A^{\prime }=U$ for $U=\{1,\dots ,20\}$, $A=\{1,3,5,\dots ,19\}$

Verify $A\cup A^{\prime }=U$.

1. $A^{\prime }=U-A=\{2,4,6,\dots ,20\}$ (the even numbers).
2. $A\cup A^{\prime }=\{1,2,\dots ,20\}=U$.

Answer: Verified: $A\cup A^{\prime }=U$

Q11(ii) — Verify $A\cap U=A$

Verify $A\cap U=A$.

1. Since $A\subseteq U$, intersecting $A$ with the universal set returns $A$ itself.

Answer: Verified: $A\cap U=A$

Q11(iii) — Verify $A\cap A^{\prime }=\varnothing$

Verify $A\cap A^{\prime }=\varnothing$.

1. $A$ and its complement $A^{\prime }$ share no elements by definition.

Answer: Verified: $A\cap A^{\prime }=\varnothing$

WP1 — Cricket and hockey (55 students)

55 students; 34 like cricket, 30 like hockey; each likes at least one. How many like both?

1. $n(C\cup H)=n(C)+n(H)-n(C\cap H)$.
2. $55=34+30-n(C\cap H)\Rightarrow n(C\cap H)=64-55$.

Answer: $9$ students like both games

WP2 — Three languages (500 employees)

500 employees; $250$ Urdu, $150$ English, $50$ Punjabi; pairwise overlaps $40,30,10$ (U∩E, E∩P, U∩P); everyone speaks at least one. Find how many speak all three.

1. $n(U\cup E\cup P)=n(U)+n(E)+n(P)-n(U\cap E)-n(E\cap P)-n(U\cap P)+n(U\cap E\cap P)$.
2. $500=250+150+50-40-30-10+x=370+x$.

Answer: $x=130$ (applying the standard three-set formula to the given data)

WP3 — Shirts and caps (34 total)

19 blue shirts, 15 green shirts, $3$ blue∩green, $4$ cap∩blue, $2$ cap∩green, no triple overlap given (assume $0$); total wearing any of the three is $34$. Find cap-wearers.

1. $34=19+15+c-3-4-2+0=25+c$.

Answer: $c=9$ people wear caps

WP4 — Laptops, tablets, books (35 total)

17 laptops, 11 tablets, $9$ laptop∩tablet, $6$ laptop∩books, $4$ tablet∩books, $8$ have all three; total with any item is $35$. Find how many have books.

1. $35=17+11+b-9-6-4+8=17+b$.

Answer: $b=18$ participants have books

WP5(a) — Shopping mall salary sets: $(A^{\prime }\cup B^{\prime })\cap C$

$U=\{1,\dots ,150\}$, $A=\{50,\dots ,89\}$, $B=\{101,\dots ,150\}$, $C=\{1,\dots ,49\}\cup \{90,\dots ,100\}$. Find $(A^{\prime }\cup B^{\prime })\cap C$.

1. $A^{\prime }=\{1,\dots ,49\}\cup \{90,\dots ,150\}$; $B^{\prime }=\{1,\dots ,100\}$.
2. $A^{\prime }\cup B^{\prime }$ covers everything from $1$–$100$ (via $B^{\prime }$) together with $90$–$150$ (via $A^{\prime }$), i.e. all of $U$.
3. $(A^{\prime }\cup B^{\prime })\cap C=U\cap C=C$.

Answer: $(A^{\prime }\cup B^{\prime })\cap C=C=\{1,\dots ,49\}\cup \{90,\dots ,100\}$ (60 elements)

WP5(b) — Find $n\{A\cap (B^{\prime }\cap C^{\prime })\}$

Using the same sets, find $n\{A\cap (B^{\prime }\cap C^{\prime })\}$.

1. $C^{\prime }=\{50,\dots ,89\}\cup \{101,\dots ,150\}$.
2. $B^{\prime }\cap C^{\prime }=\{1,\dots ,100\}\cap (\{50,\dots ,89\}\cup \{101,\dots ,150\})=\{50,\dots ,89\}$.
3. $A\cap (B^{\prime }\cap C^{\prime })=\{50,\dots ,89\}\cap \{50,\dots ,89\}=A$, which has $89-50+1$ elements.

Answer: $n\{A\cap (B^{\prime }\cap C^{\prime })\}=40$

WP6(a) — Cricket, football, hockey (125 students)

60 cricket, 70 football, 40 hockey; pairwise overlaps $25,15,10$; total (each plays at least one) is $125$. Find how many play all three.

1. $125=60+70+40-25-15-10+x=120+x$.

Answer: $x=5$ students play all three sports

WP6(b) — Venn diagram for the sports problem

Draw a Venn diagram showing the distribution.

1. Only cricket $=60-[(25-5)+(10-5)+5]=30$.
2. Only football $=70-[(25-5)+(15-5)+5]=35$.
3. Only hockey $=40-[(15-5)+(10-5)+5]=20$.
4. Cricket∩football only $=20$; football∩hockey only $=10$; cricket∩hockey only $=5$; all three $=5$.
5. Check: $30+35+20+20+10+5+5=125$ ✓

Answer: Regions: only-C$=30$, only-F$=35$, only-H$=20$, C∩F$=20$, F∩H$=10$, C∩H$=5$, all three$=5$

WP7(a) — Nihari, biryani, korma survey (130 people): at least one

40 nihari, 65 biryani, 50 korma; pairwise overlaps $20,35,27$; all three $=12$. At least how many like nihari, biryani, or korma?

1. $n(N\cup B\cup K)=40+65+50-20-35-27+12=155-82+12$.

Answer: $85$ people

WP7(b) — How many like none of the three?

Using the same data, how many like none of the three foods?

1. Total surveyed $-$ those who like at least one: $130-85$.

Answer: $45$ people

WP7(c) — How many like only one food?

Using the same data, how many like exactly one of the three foods?

1. Only nihari $=40-20-27+12=5$.
2. Only biryani $=65-20-35+12=22$.
3. Only korma $=50-27-35+12=0$.
4. Sum $=5+22+0$.

Answer: $27$ people like exactly one food

WP7(d) — Venn diagram for the food survey

Draw a Venn diagram for the food survey.

1. Only-N$=5$, only-B$=22$, only-K$=0$, N∩B only$=8$, B∩K only$=23$, N∩K only$=15$, all three$=12$.
2. Check: $5+22+0+8+23+15+12=85$ (inside the three circles), plus $45$ outside $=130$ ✓

Answer: Regions as listed; $45$ people lie outside all three circles

Q12(i) — Relation $\{(x,y)\mid y=x\}$

Find the relation in $A=\{1,2,3,4\}$ and its domain/range.

1. Pairs where $y=x$: $(1,1),(2,2),(3,3),(4,4)$.

Answer: Relation $=\{(1,1),(2,2),(3,3),(4,4)\}$; Domain $=$ Range $=\{1,2,3,4\}$

Q12(ii) — Relation $\{(x,y)\mid y+x=5\}$

Find the relation and its domain/range.

1. Pairs with $x+y=5$: $(1,4),(2,3),(3,2),(4,1)$.

Answer: Relation $=\{(1,4),(2,3),(3,2),(4,1)\}$; Domain $=$ Range $=\{1,2,3,4\}$

Q12(iii) — Relation $\{(x,y)\mid x+y<5\}$

Find the relation and its domain/range.

1. Checking each $x$: $x=1\Rightarrow y\in \{1,2,3\}$; $x=2\Rightarrow y\in \{1,2\}$; $x=3\Rightarrow y=1$; $x=4$: none.

Answer: Relation $=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\}$; Domain $=\{1,2,3\}$, Range $=\{1,2,3\}$

Q12(iv) — Relation $\{(x,y)\mid x+y>5\}$

Find the relation and its domain/range.

1. Checking each $x$: $x=2\Rightarrow y=4$; $x=3\Rightarrow y\in \{3,4\}$; $x=4\Rightarrow y\in \{2,3,4\}$.

Answer: Relation $=\{(2,4),(3,3),(3,4),(4,2),(4,3),(4,4)\}$; Domain $=\{2,3,4\}$, Range $=\{2,3,4\}$

Q13(i) — $g(0)$ for $g(x)=3x+2$

Find $g(0)$.

1. $g(0)=3(0)+2$.

Answer: $2$

Q13(ii) — $g(-3)$

Find $g(-3)$.

1. $g(-3)=3(-3)+2=-9+2$.

Answer: $-7$

Q13(iii) — $g\left(\frac{2}{3}\right)$

Find $g\left(\frac{2}{3}\right)$.

1. $g\left(\frac{2}{3}\right)=3\left(\frac{2}{3}\right)+2=2+2$.

Answer: $4$

Q13(iv) — $h(1)$ for $h(x)=x^2+1$

Find $h(1)$.

1. $h(1)=1^2+1$.

Answer: $2$

Q13(v) — $h(-4)$

Find $h(-4)$.

1. $h(-4)=(-4{)}^2+1=16+1$.

Answer: $17$

Q13(vi) — $h\left(-\frac{1}{2}\right)$

Find $h\left(-\frac{1}{2}\right)$.

1. $h\left(-\frac{1}{2}\right)={\left(-\frac{1}{2}\right)}^2+1=\frac{1}{4}+1$.

Answer: $\frac{5}{4}$

Q14(i) — Find $a,b$: $f(x)=ax+b+1$, $f(3)=8$, $f(6)=14$

Find the values of $a$ and $b$.

1. $f(3)=8\Rightarrow 3a+b=7$.
2. $f(6)=14\Rightarrow 6a+b=13$.
3. Subtract: $3a=6\Rightarrow a=2$; then $b=7-3(2)=1$.

Answer: $a=2,b=1$

Q14(ii) — Find $a,b$: $g(x)=ax+b+5$, $g(-1)=0$, $g(2)=10$

Find the values of $a$ and $b$.

1. $g(-1)=0\Rightarrow -a+b=-5$.
2. $g(2)=10\Rightarrow 2a+b=5$.
3. Subtract: $3a=10\Rightarrow a=\frac{10}{3}$; then $b=-5+a=-\frac{5}{3}$.

Answer: $a=\frac{10}{3},b=-\frac{5}{3}$

Q14(iii) — $f(x)=5x+2$, $f(x)=32$, find $x$

Find $x$ if $f(x)=32$.

1. $5x+2=32\Rightarrow 5x=30$.

Answer: $x=6$

Q14(iv) — Find $c,d$: $f(x)=c{x}^2+d$, $f(1)=6$, $f(-2)=10$

Find the values of $c$ and $d$.

1. $f(1)=6\Rightarrow c+d=6$.
2. $f(-2)=10\Rightarrow 4c+d=10$.
3. Subtract: $3c=4\Rightarrow c=\frac{4}{3}$; then $d=6-\frac{4}{3}=\frac{14}{3}$.

Answer: $c=\frac{4}{3},d=\frac{14}{3}$

Q15(i) — Tabular: $\{x\mid x=2n,n\in N\}$

Write in tabular form.

1. $n=1,2,3,\dots$ gives all even numbers.

Answer: $\{2,4,6,8,\dots \}$

Q15(ii) — Tabular: $\{x\mid x=2m+1,m\in N\}$

Write in tabular form.

1. $m=1,2,3,\dots$ gives $3,5,7,\dots$

Answer: $\{3,5,7,9,\dots \}$

Q15(iii) — Tabular: $\{x\mid x=11n,n\in W\wedge n<11\}$

Write in tabular form.

1. $W=\{0,1,2,\dots \}$; $n=0,\dots ,10$.

Answer: $\{0,11,22,33,44,55,66,77,88,99,110\}$

Q15(iv) — Tabular: $\{x\mid x\in E\wedge 4<x<6\}$

Write in tabular form ($E$ = even numbers).

1. Strictly between $4$ and $6$ is only $5$, which is odd — no even number qualifies.

Answer: $\{\}=\varnothing$

Q15(v) — Tabular: $\{x\mid x\in O\wedge 5<x<7\}$

Write in tabular form ($O$ = odd numbers).

1. Strictly between $5$ and $7$ is only $6$, which is even — no odd number qualifies.

Answer: $\{\}=\varnothing$

Q15(vi) — Tabular: $\{x\mid x\in Q\wedge x^2=2\}$

Write in tabular form.

1. $x=\pm \sqrt{2}$, which is irrational, not in $Q$.

Answer: $\{\}=\varnothing$

Q15(vii) — Tabular: $\{x\mid x\in Q\wedge x=-x\}$

Write in tabular form.

1. $x=-x\Rightarrow 2x=0\Rightarrow x=0$, and $0\in Q$.

Answer: $\{0\}$

Q15(viii) — $\{x\mid x\in R\wedge x\notin Q^{\prime }\}$

Describe this set.

1. $Q^{\prime }$ is the set of irrationals; "not irrational" among the reals means rational.
2. So this set is exactly $Q$, which is infinite and cannot be listed in tabular form.

Answer: This set equals $Q$ (the rational numbers) — infinite, no tabular listing

Q16(i) — Find $A^{\prime }$

List $A^{\prime }$.

1. $A^{\prime }=U-A$.

Answer: $\{1,3,5,7,9\}$

Q16(ii) — Find $B^{\prime }$

List $B^{\prime }$.

1. $B^{\prime }=U-B$.

Answer: $\{6,7,8,9,10\}$

Q16(iii) — Find $A\cup B$

List $A\cup B$.

1. Combine both, removing duplicates ($2,4$ appear in both).

Answer: $\{1,2,3,4,5,6,8,10\}$

Q16(iv) — Find $A-B$

List $A-B$.

1. Elements of $A$ that are not in $B$.

Answer: $\{6,8,10\}$

Q16(v) — Find $A\cap C$

List $A\cap C$.

1. $A$ is all even, $C$ is all odd — no overlap.

Answer: $\{\}=\varnothing$

Q16(vi) — Find $A^{\prime }\cup C^{\prime }$

List $A^{\prime }\cup C^{\prime }$.

1. $A^{\prime }=\{1,3,5,7,9\}$, $C^{\prime }=U-C=\{2,4,6,8,10\}$; their union covers everything in $U$.

Answer: $\{1,2,3,4,5,6,7,8,9,10\}=U$

Q16(vii) — Find $A^{\prime }\cap C$

List $A^{\prime }\cap C$.

1. $A^{\prime }=\{1,3,5,7,9\}$, which is identical to $C=\{1,3,5,7,9\}$.

Answer: $\{1,3,5,7,9\}$

Q16(viii) — Find $U^{\prime }$

List $U^{\prime }$.

1. The complement of the universal set (within itself) is always empty.

Answer: $\{\}=\varnothing$

Q17(i) — Simplify $A^{\prime }$

Find the single set equal to $A^{\prime }$.

1. $A^{\prime }$ is already a single-set expression; it cannot be simplified further without knowing $A$ explicitly.

Answer: $A^{\prime }$ (already in simplest form)

Q17(ii) — Simplify $A\cap U$

Find the single set equal to $A\cap U$.

1. Intersecting any set with the universal set returns that set.

Answer: $A$

Q17(iii) — Simplify $A\cup U$

Find the single set equal to $A\cup U$.

1. Union with the universal set always gives the universal set.

Answer: $U$

Q17(iv) — Simplify $A\cup \varnothing$

Find the single set equal to $A\cup \varnothing$.

1. Identity law: union with the empty set changes nothing.

Answer: $A$

Q17(v) — Simplify $\varnothing \cap \varnothing$

Find the single set equal to $\varnothing \cap \varnothing$.

1. The empty set intersected with itself is still empty.

Answer: $\varnothing$

Q18(i) — Verify $A-B=A\cap B^{\prime }$

Use a Venn diagram to verify the difference-of-sets identity.

1. Shade $A$, then remove the part overlapping $B$ — what remains is exactly $A$ intersected with the region outside $B$ (i.e. $B^{\prime }$).
2. Example check: $U=\{1,2,3,4\}$, $A=\{1,2\}$, $B=\{2,3\}$. $A-B=\{1\}$. $B^{\prime }=\{1,4\}$, so $A\cap B^{\prime }=\{1,2\}\cap \{1,4\}=\{1\}$ — matches.

Answer: Verified: $A-B=A\cap B^{\prime }$

Q18(ii) — Verify $(A-B{)}^{\prime }\cap B=B$

Use a Venn diagram to verify this identity.

1. $(A-B{)}^{\prime }=(A\cap B^{\prime }{)}^{\prime }=A^{\prime }\cup B$ by De Morgan's law.
2. $(A^{\prime }\cup B)\cap B=(A^{\prime }\cap B)\cup (B\cap B)=(A^{\prime }\cap B)\cup B=B$, since $A^{\prime }\cap B\subseteq B$.
3. Example check: $U=\{1,2,3,4\}$, $A=\{1,2\}$, $B=\{2,3\}$: $A-B=\{1\}$, $(A-B{)}^{\prime }=\{2,3,4\}$, $(A-B{)}^{\prime }\cap B=\{2,3,4\}\cap \{2,3\}=\{2,3\}=B$ — matches.

Answer: Verified: $(A-B{)}^{\prime }\cap B=B$

Q19(a) — Verify all four laws for $A=\{1,2,3,4\}$, $B=\{3,4,5,6,7,8\}$, $C=\{5,6,7,9,10\}$

Verify associativity of union/intersection and distributivity both ways.

1. Union associativity: $(A\cup B)\cup C=\{1,\dots ,8\}\cup C=\{1,\dots ,10\}$; $A\cup (B\cup C)=A\cup \{3,\dots ,10\}=\{1,\dots ,10\}$ — same.
2. Intersection associativity: $(A\cap B)\cap C=\{3,4\}\cap C=\varnothing$; $A\cap (B\cap C)=A\cap \{5,6,7\}=\varnothing$ — same.
3. Union over intersection: $A\cup (B\cap C)=A\cup \{5,6,7\}=\{1,2,3,4,5,6,7\}$; $(A\cup B)\cap (A\cup C)=\{1,\dots ,8\}\cap \{1,2,3,4,5,6,7,9,10\}=\{1,\dots ,7\}$ — same.
4. Intersection over union: $A\cap (B\cup C)=A\cap \{3,\dots ,10\}=\{3,4\}$; $(A\cap B)\cup (A\cap C)=\{3,4\}\cup \varnothing =\{3,4\}$ — same.