📘 Class 9 (Maharashtra Board) — Chapter 1: Sets
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Part A • 1-Mark Questions (20) — Red Q / Green A
1) What is a set?
A well-defined collection of distinct objects (elements).
2) Symbol for “\(a\) is an element of set \(A\)”.
\(\;a \in A\; \) (and \(a \notin A\) means \(a\) is not an element).
3) Write the set of natural numbers \(N\).
\(N=\{1,2,3,\dots\}\).
4) Write the set of whole numbers \(W\).
\(W=\{0,1,2,3,\dots\}\).
5) List the letters of “remember” as a set.
\(\{r,e,m,b\}\) (each written once; order doesn’t matter).
6) Roster form of odd numbers between 1 and 10.
\(\{3,5,7,9\}\).
7) Builder form for primes between 1 and 10.
\(\{x \mid x \text{ is prime},\,1
8) What is a singleton set? Give an example.
A set with exactly one element; e.g. \(\{2\}\).
9) Symbol for the empty set.
\(\varnothing\) or \(\{\}\).
10) Decide: \(\{ \text{Happy children in the village}\}\) — set or not?
Not a set (not well-defined; “happy” is relative).
11) Equal sets \(A\) and \(B\) mean?
Every element of \(A\) is in \(B\) and every element of \(B\) is in \(A\) (write \(A=B\)).
12) Define subset.
\(B\subseteq A\) if every element of \(B\) is an element of \(A\).
13) Write the complement of \(A\) in universal set \(U\).
\(A^{\prime}=\{x\mid x\in U,\;x\notin A\}\).
14) Define intersection.
\(A\cap B=\{x\mid x\in A \text{ and } x\in B\}\).
15) Define union.
\(A\cup B=\{x\mid x\in A \text{ or } x\in B\}\).
16) What are disjoint sets?
Sets with no common elements; \(A\cap B=\varnothing\).
17) If \(B\subseteq A\), then \(A\cap B=\ ?\)
\(A\cap B=B\).
18) Write the counting rule for union.
\(\;n(A\cup B)=n(A)+n(B)-n(A\cap B)\;.\)
19) Complement of \(U\).
\(U'=\varnothing\).
20) \((A')'=\ ?\)
\((A')'=A\).
Part B • 2-Mark Questions (20) — Red Q / Green A
1) Write in roster form and classify as finite/infinite: \(A=\{x\mid x\in \mathbb{N},\, x\text{ is odd}\}\).
\(\{1,3,5,7,\dots\}\); Infinite.
2) Decide if \(\{ \text{Days of a week}\}\) is a set. Give roster form.
Yes. \(\{\text{Sunday},\text{Monday},\dots,\text{Saturday}\}\).
3) Convert to builder form: \(B=\{-3,3\}\).
\(\;B=\{y\mid y^2=9\}\).
4) Convert to roster: \(C=\{z\mid z\text{ is a multiple of }5,\;z<30\}\).
\(\{5,10,15,20,25\}\).
5) If \(N=\mathbb{N},\ I=\mathbb{Z}\), show \(N\subseteq I\).
Every natural number is an integer, hence \(N\subseteq I\).
6) List all elements: \(P=\{x\mid x=\sqrt{25}\}\), \(S=\{y\mid y\in\mathbb{Z},-5\le y\le 5\}\). Decide \(P\subseteq S\).
\(P=\{-5,5\}\), \(S=\{-5,-4,-3,-2,-1,0,1,2,3,4,5\}\); so \(P\subseteq S\).
7) Are \(A=\{1,3,5,7\}\) and \(D=\{2,3,5,7\}\) equal?
No. \(1\in A\) but \(1\notin D\).
8) Write complement: In \(U=\{1,\dots,10\}\), for \(A=\{2,4,6,8,10\}\) find \(A'\).
\(A'=\{1,3,5,7,9\}\).
9) If \(U=\{1,3,9,11,13,18,19\}\), \(B=\{3,9,11,13\}\), find \(B'\).
\(B'=\{1,18,19\}\).
10) Prove: If \(B\subseteq A\), then \(A\cap B=B\).
Since all elements of \(B\) are in \(A\), the common elements are exactly those in \(B\); hence \(A\cap B=B\).
11) Find \(A\cup B\) for \(A=\{-1,-3,-5,0\}\), \(B=\{0,3,5\}\).
\(A\cup B=\{-5,-3,-1,0,3,5\}\).
12) State two properties of complement.
\(U'=\varnothing\), \(\varnothing'=U\), \(A\cap A'=\varnothing\), \(A\cup A'=U\), \((A')'=A\).
13) Determine if \(\{ \text{Brave children in the class}\}\) is a set.
Not a set (brave is not well-defined).
14) Give builder form for \(Q\) (rationals).
\(\displaystyle Q=\left\{\frac{p}{q}\ \middle|\ p,q\in\mathbb{Z},\ q\neq 0\right\}.\)
15) Write \(X=\{1,4,9,\dots,100\}\) in builder form.
\(X=\{n^2\mid n\in \mathbb{N},\,1\le n\le 10\}\).
16) Decide: \(\{x\mid x^2=100,\ x\in\mathbb{Z}\}\): finite or infinite?
Finite; \(=\{-10,10\}\).
17) Find \(n(A\cup B)\) if \(n(A)=7\), \(n(B)=13\), \(n(A\cap B)=4\).
\(n(A\cup B)=7+13-4=16\).
18) If \(A\subseteq B\), compare \(A\cup B\) with \(B\).
\(A\cup B=B\).
19) Write \((A\cup B)'\) in terms of \(A',B'\).
\((A\cup B)'=A'\cap B'\) (De Morgan’s law).
20) Write \((A\cap B)'\) in terms of \(A',B'\).
\((A\cap B)'=A'\cup B'\).
Part C • 3-Mark Questions (20) — Red Q / Green A
1) Show that \(n(A)+n(B)=n(A\cup B)+n(A\cap B)\).
From \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\), rearrange to get \(n(A)+n(B)=n(A\cup B)+n(A\cap B)\).
2) In a class of 70, \(n(A)=45\) (Cricket), \(n(B)=52\) (Kho-Kho), all like at least one. Find \(n(A\cap B)\), only-Cricket, only-Kho-Kho.
\(n(A\cup B)=70\). So \(n(A\cap B)=45+52-70=27\). Only-Cricket \(=45-27=18\). Only-Kho-Kho \(=52-27=25\).
3) If \(B\subseteq A\), prove \(A\cap B=B\) and \(A\cup B=A\).
By definition of subset, common elements are \(B\) itself; union adds nothing beyond \(A\).
4) For \(U=\{1,\dots,12\}\), \(A=\{2,4,6,8,10\}\), \(B=\{1,3,5,7,8,10\}\). Find \(A\cup B\), \(A\cap B\), \(A'\), \(B'\), \((A\cup B)'\), \((A\cap B)'\).
\(A\cup B=\{1,2,3,4,5,6,7,8,10\}\); \(A\cap B=\{8,10\}\); \(A'=\{1,3,5,7,9,11,12\}\); \(B'=\{2,4,6,9,11,12\}\); \((A\cup B)'=\{9,11,12\}\); \((A\cap B)'=\{1,2,3,4,5,6,7,9,11,12\}\).
5) Fill: If \(n(A)=15\), \(n(A\cup B)=29\), \(n(A\cap B)=7\), find \(n(B)\).
\(29=15+n(B)-7\Rightarrow n(B)=21\).
6) Hostel: 125 students; tea 80, coffee 60, both 20. How many drink neither?
\(n(A\cup B)=80+60-20=120\). Neither \(=125-120=5\).
7) Competitive exam: Eng 50, Math 60, both 40, none fail in both. Find passed at least one.
At least one \(=50+60-40=70\).
8) Survey: 220 students; rock 130, sky 180, both 110. Find neither, rock-only, sky-only.
Union \(=130+180-110=200\). Neither \(=220-200=20\). Rock-only \(=130-110=20\). Sky-only \(=180-110=70\).
9) Decide finite/infinite: \(B=\{y\mid y<-1,\ y\in\mathbb{Z}\}\); \(W\) (whole numbers).
\(B=\{-2,-3,-4,\dots\}\) infinite; \(W=\{0,1,2,\dots\}\) infinite.
10) If \(P\subseteq M\), compute \(P\cap(P\cup M)\).
\(P\cup M=M\); so \(P\cap M=P\).
11) Write all ordered pairs \((a,b)\in W\times W\) with \(a+b=9\).
\((0,9),(1,8),(2,7),(3,6),(4,5),(5,4),(6,3),(7,2),(8,1),(9,0)\).
12) If \(A=\{1,2,3,5,7,9,11,13\}\), \(B=\{1,2,4,6,8,12,13\}\), verify \(n(A)+n(B)=n(A\cup B)+n(A\cap B)\).
\(n(A)=8\), \(n(B)=7\), \(A\cap B=\{1,2,13\}\Rightarrow 3\), \(A\cup B\) has 12 distinct elements. \(8+7=12+3=15\) ✓.
13) Show using De Morgan: \((A\cup B)'=A'\cap B'\).
An element is outside \(A\cup B\) iff it is outside both \(A\) and \(B\); thus in \(A'\cap B'\).
14) Decide equality: \(A=\{x\mid x=2n,\ n\in\mathbb{N},\ 0
Both \(\{2,4,6,8,10\}\); hence \(A=B\).
15) Write five different subsets of \(A=\{1,3,4,7,8\}\).
\(\{1,3\},\{4,7,8\},\{1,4,8\},\{1,4,7,8\},\{3,8\}\) (many more possible).
16) With universal set “students of class 9”, describe \(A'\) when \(A\) = students scoring \(\ge 50\%\) in Maths.
\(A'=\) students scoring < \(50\%\) in Maths.
17) Prove: \(A\cap A'=\varnothing\) and \(A\cup A'=U\).
No element can be simultaneously in \(A\) and not in \(A\); every element is either in \(A\) or not in \(A\).
18) Show that if \(A\cap B=B\), then \(B\subseteq A\).
Every \(x\in B\) lies in \(A\cap B\), hence \(x\in A\); so \(B\subseteq A\).
19) List \(N\cap W\).
\(N\cap W=N=\{1,2,3,\dots\}\) (assuming \(N\) starts at 1).
20) For \(T=\{1,2,3,4,5\}\), \(M=\{3,4,7,8\}\), find \(T\cup M\) and \(T\cap M\).
\(T\cup M=\{1,2,3,4,5,7,8\}\); \(T\cap M=\{3,4\}\).
Part D • Textbook Exercises — Perfect Solutions (MathJax used)
Practice Set 1.1 — Solutions
(1) Write the following sets in roster form: (i) even numbers (ii) even primes 1–50 (iii) negative integers (iv) seven basic sounds of sargam.
(i) \(\{2,4,6,8,\dots\}\).
(ii) \(\{2\}\).
(iii) \(\{-1,-2,-3,\dots\}\).
(iv) \(\{\text{Sa},\text{Re},\text{Ga},\text{Ma},\text{Pa},\text{Dha},\text{Ni}\}\).
(ii) \(\{2\}\).
(iii) \(\{-1,-2,-3,\dots\}\).
(iv) \(\{\text{Sa},\text{Re},\text{Ga},\text{Ma},\text{Pa},\text{Dha},\text{Ni}\}\).
(2) Write symbolic statements in words: (i) \(\frac{4}{3}\in Q\) (ii) \(-2\notin N\) (iii) \(P=\{p\mid p\text{ is odd}\}\).
(i) “Four by three is a rational number.”
(ii) “Minus two is not a natural number.”
(iii) “Set \(P\) is the set of all odd numbers.”
(ii) “Minus two is not a natural number.”
(iii) “Set \(P\) is the set of all odd numbers.”
(3) Write any two sets by listing method AND rule method.
Example 1: Roster \(\{2,4,6,8\}\); Rule \(\{x\mid x=2n,\ 1\le n\le 4\}\).
Example 2: Roster \(\{a,e,i,o,u\}\); Rule \(\{x\mid x\text{ is a vowel}\}\).
Example 2: Roster \(\{a,e,i,o,u\}\); Rule \(\{x\mid x\text{ is a vowel}\}\).
(4) Listing method: (i) All months in the Indian solar year (ii) Letters in “COMPLEMENT” (iii) Human sensory organs (iv) Primes 1–20 (v) Continents.
(i) \(\{\text{Chaitra},\text{Vaishakha},\text{Jyeshtha},\text{Ashadha},\text{Shravana},\text{Bhadrapada},\text{Ashwin},\text{Kartika},\text{Margashirsha},\text{Pausha},\text{Magha},\text{Phalguna}\}\).
(ii) \(\{C,O,M,P,L,E,N,T\}\).
(iii) \(\{\text{Eyes},\text{Ears},\text{Nose},\text{Tongue},\text{Skin}\}\).
(iv) \(\{2,3,5,7,11,13,17,19\}\).
(v) \(\{\text{Asia},\text{Africa},\text{North America},\text{South America},\text{Antarctica},\text{Europe},\text{Australia}\}\).
(ii) \(\{C,O,M,P,L,E,N,T\}\).
(iii) \(\{\text{Eyes},\text{Ears},\text{Nose},\text{Tongue},\text{Skin}\}\).
(iv) \(\{2,3,5,7,11,13,17,19\}\).
(v) \(\{\text{Asia},\text{Africa},\text{North America},\text{South America},\text{Antarctica},\text{Europe},\text{Australia}\}\).
(5) Rule method: (i) \(A=\{1,4,9,\dots,100\}\) (ii) \(B=\{6,12,18,\dots,48\}\) (iii) \(C=\{S,M,I,L,E\}\) (iv) Days of week \(D\) (v) \(X=\{a,e,t\}\).
(i) \(A=\{n^2\mid n\in\mathbb{N},\ 1\le n\le 10\}\).
(ii) \(B=\{6n\mid n\in\mathbb{N},\ 1\le n\le 8\}\).
(iii) \(C=\{x\mid x\text{ is a letter of “SMILE”}\}\).
(iv) \(D=\{x\mid x\text{ is a day of the week}\}\).
(v) \(X=\{x\mid x\in\{a,e,t\}\}\).
(ii) \(B=\{6n\mid n\in\mathbb{N},\ 1\le n\le 8\}\).
(iii) \(C=\{x\mid x\text{ is a letter of “SMILE”}\}\).
(iv) \(D=\{x\mid x\text{ is a day of the week}\}\).
(v) \(X=\{x\mid x\in\{a,e,t\}\}\).
Types of Sets — Worked Examples
Ex.: Write in roster and classify finite/infinite: (i) \(A=\{x\in\mathbb{N}\mid x\text{ odd}\}\) (ii) \(B=\{x\in\mathbb{N}\mid 3x-1=0\}\) (iii) \(C=\{x\in\mathbb{N}\mid 7\mid x\}\) (iv) \(D=\{(a,b)\in W^2\mid a+b=9\}\) (v) \(E=\{x\in\mathbb{Z}\mid x^2=100\}\) (vi) \(F=\{(a,b)\in Q^2\mid a+b=11\}\).
(i) \(\{1,3,5,7,\dots\}\) Infinite.
(ii) \(3x=1\Rightarrow x=\frac{1}{3}\notin\mathbb{N}\Rightarrow B=\varnothing\) (finite).
(iii) \(\{7,14,21,\dots\}\) Infinite.
(iv) \(\{(0,9),(1,8),\dots,(9,0)\}\) Finite (10 pairs).
(v) \(\{-10,10\}\) Finite.
(vi) Infinitely many rational pairs, so Infinite.
(ii) \(3x=1\Rightarrow x=\frac{1}{3}\notin\mathbb{N}\Rightarrow B=\varnothing\) (finite).
(iii) \(\{7,14,21,\dots\}\) Infinite.
(iv) \(\{(0,9),(1,8),\dots,(9,0)\}\) Finite (10 pairs).
(v) \(\{-10,10\}\) Finite.
(vi) Infinitely many rational pairs, so Infinite.
Practice Set 1.2 — Solutions
(1) Decide which are equal: \(A=\{x\mid 3x-1=2\}\), \(B=\{x\mid x\in\mathbb{N} \text{ but } x\text{ neither prime nor composite}\}\), \(C=\{x\in\mathbb{N}\mid x<2\}\).
\(A:\ 3x=3\Rightarrow x=1\Rightarrow A=\{1\}\).
\(B:\) Only \(1\) is neither prime nor composite \(\Rightarrow B=\{1\}\).
\(C:\ \{1\}\) (since natural numbers <2 is just 1). Hence \(A=B=C\) (equal).
\(B:\) Only \(1\) is neither prime nor composite \(\Rightarrow B=\{1\}\).
\(C:\ \{1\}\) (since natural numbers <2 is just 1). Hence \(A=B=C\) (equal).
(2) Decide equality: \(A=\{\text{even prime numbers}\}\), \(B=\{x\mid 7x-1=13\}\).
\(A=\{2\}\). For \(B\): \(7x=14\Rightarrow x=2\Rightarrow B=\{2\}\). Thus \(A=B\).
(3) Which are empty sets? Why? (i) \(A=\{a\mid a\) is a natural number smaller than 0\(\}\) (ii) \(B=\{x\mid x^2=0\}\) (iii) \(C=\{x\mid 5x-2=0,\ x\in\mathbb{N}\}\).
(i) \(\varnothing\) (no natural numbers <0).
(ii) Not empty; \(x=0\Rightarrow B=\{0\}\).
(iii) \(5x=2\Rightarrow x=\frac{2}{5}\notin\mathbb{N}\Rightarrow \varnothing\).
(ii) Not empty; \(x=0\Rightarrow B=\{0\}\).
(iii) \(5x=2\Rightarrow x=\frac{2}{5}\notin\mathbb{N}\Rightarrow \varnothing\).
(4) Write with reasons: finite/infinite — (i) \(A=\{x\mid x<10,\ x\in\mathbb{N}\}\) (ii) \(B=\{y\mid y<-1,\ y\in\mathbb{Z}\}\) (iii) \(C=\) students of class 9 in your school (iv) people from your village (v) lab apparatus (vi) whole numbers (vii) rationals.
(i) Finite \(\{1,\dots,9\}\). (ii) Infinite. (iii) Finite. (iv) Finite (though large). (v) Finite. (vi) Infinite. (vii) Infinite.
Practice Set 1.3 — Solutions
(1) With \(A=\{a,b,c,d,e\}\), \(B=\{c,d,e,f\}\), \(C=\{b,d\}\), \(D=\{a,e\}\), decide: (i) \(C\subseteq B\) (ii) \(A\subseteq D\) (iii) \(D\subseteq B\) (iv) \(D\subseteq A\) (v) \(B\subseteq A\) (vi) \(C\subseteq A\).
(i) True (b? wait, \(C=\{b,d\}\) — \(b\notin B\)) ⇒ False.
(ii) False. (iii) False (\(a\notin B\)). (iv) True. (v) False (f∉A). (vi) True.
(ii) False. (iii) False (\(a\notin B\)). (iv) True. (v) False (f∉A). (vi) True.
(2) Take \(U=\{1,\dots,20\}\), show \(X=\{x\in\mathbb{N}\mid 7<x<15\}\), \(Y=\{y\in\mathbb{N}\mid y\) prime from 1 to 20\(\}\).
\(X=\{8,9,10,11,12,13,14\}\). \(Y=\{2,3,5,7,11,13,17,19\}\).
(3) \(U=\{1,2,3,7,8,9,10,11,12\}\), \(P=\{1,3,7,10\}\). Show \(U,P,P'\) and verify \((P')'=P\).
\(P'=\{2,8,9,11,12\}\). Then \((P')'=P\) ✓.
(4) For \(A=\{1,3,2,7\}\), write any three subsets.
\(\{1,3\}\), \(\{2,7\}\), \(\{1,2,7\}\).
(5) Residents: \(P\) Pune, \(M\) Madhya Pradesh, \(I\) Indore, \(B\) India, \(H\) Maharashtra. Write subset relations and a universal set.
\(I\subseteq M\subseteq B\), \(P\subseteq H\subseteq B\). Universal set can be \(B=\) residents of India.
(6*) Choose a universal set for: (i) multiples of 5, 7, 12 (ii) integers multiples of 4 and even squares.
(i) \(U=\mathbb{N}\) or \(\mathbb{Z}\). (ii) \(U=\mathbb{Z}\) (or \(\mathbb{N}\) if restricting to non-negative).
(7) Let \(U\) be all students; \(A=\) students with \(\ge 50\%\) in Maths. Write \(A'\).
\(A'=\) students with < \(50\%\) in Maths.
Practice Set 1.4 — Solutions
(1) If \(n(A)=15\), \(n(A\cup B)=29\), \(n(A\cap B)=7\), find \(n(B)\).
\(\;29=15+n(B)-7 \Rightarrow n(B)=21\).
(2) Hostel: 125 students; tea 80, coffee 60, both 20. Find who drink neither.
Neither \(=125-(80+60-20)=5\).
(3) Exam: Eng 50, Math 60, both 40, none fail both. Number passed at least one?
At least one \(=50+60-40=70\).
(4*) Survey: 220 students; rock 130, sky 180, both 110. Find neither, rock-only, sky-only.
Union \(=200\); Neither \(=20\); Rock-only \(=20\); Sky-only \(=70\).
(5) From Venn diagram with regions \(p,q,r,s,m,n,y,z\) in \(U\): write \(A,B,A\cup B,U,A',B',(A\cup B)'\).
\(A\): all regions in circle \(A\): \(\{p,q,r\}\) and shared \(\{m\}\) (labels as per diagram).
\(B\): \(\{m,n,t\}\) etc. Since labels vary by print, the rule is: take the named regions inside each set. Complement = all regions in \(U\) not in that set.
\(B\): \(\{m,n,t\}\) etc. Since labels vary by print, the rule is: take the named regions inside each set. Complement = all regions in \(U\) not in that set.
Problem Set 1 — Solutions
(1)(i) If \(M=\{1,3,5\}\), \(N=\{2,4,6\}\), find \(M\cap N\).
\(\varnothing\).
(1)(ii) \(P=\{x\mid x\) odd natural, \(1<x\le 5\}\). Roster form?
\(\{3,5\}\).
(1)(iii) \(P=\{1,2,\dots,10\}\). Type?
Finite set.
(1)(iv) If \(M\cup N=\{1,2,3,4,5,6\}\) and \(M=\{1,2,4\}\), which is \(N\)?
\(\{3,5,6\}\).
(1)(v) If \(P\subseteq M\), then \(P\cap(P\cup M)=\ ?\)
\(P\).
(1)(vi) Which are empty? (A) intersection points of parallel lines (B) even prime numbers (C) month having <30 days (D) \(P=\{x\mid x\in\mathbb{Z},-1<x<1\}\).
(A) Empty. (B) Not empty (\(\{2\}\)). (C) Empty (none). (D) Empty (no integer strictly between \(-1\) and \(1\)).
(2)(i) Which collection is a set? (A) colors of the rainbow (B) tall trees in campus (C) rich people in village (D) easy examples in book
(A) is a set (well-defined). Others are not well-defined.
(2)(ii) Which represents \(N\cap W\)?
\(\{1,2,3,\dots\}\).
(2)(iii) If \(P=\{x\mid x\) is a letter of “indian”\(\}\), roster?
\(\{i,n,d,a\}\).
(2)(iv) If \(T=\{1,2,3,4,5\}\), \(M=\{3,4,7,8\}\), find \(T\cup M\).
\(\{1,2,3,4,5,7,8\}\).
(3) Out of 100 people: 72 speak English, 43 French, each speaks at least one. Find only-English, only-French, both.
Both \(=72+43-100=15\). Only-English \(=72-15=57\). Only-French \(=43-15=28\).
(4) Trees: Parth 70, Pradnya 90, both 25. How many planted by Parth or Pradnya?
\(70+90-25=135\).
(5) If \(n(A)=20\), \(n(B)=28\), \(n(A\cup B)=36\), find \(n(A\cap B)\).
\(20+28-36=12\).
(6) In a class of 28: dog 8, cat 6, both 10. How many have neither?
Union \(=8+6-10=4\). Neither \(=28-4=24\).
(7) Represent unions with Venn diagrams & also compute: (i) \(A=\{3,4,5,7\}, B=\{1,4,8\}\) (ii) \(P=\{a,b,c,e,f\}, Q=\{l,m,n,e,b\}\) (iii) \(X=\{\text{primes } 80\text{–}100\}, Y=\{\text{odd } 90\text{–}100\}\).
(i) \(A\cup B=\{1,3,4,5,7,8\}\); \(A\cap B=\{4\}\).
(ii) \(P\cup Q=\{a,b,c,e,f,l,m,n\}\); \(P\cap Q=\{b,e\}\).
(iii) Primes 80–100 are \(\{83,89,97\}\); odd 90–100 are \(\{91,93,95,97,99\}\). So \(X\cup Y=\{83,89,91,93,95,97,99\}\); \(X\cap Y=\{97\}\).
(ii) \(P\cup Q=\{a,b,c,e,f,l,m,n\}\); \(P\cap Q=\{b,e\}\).
(iii) Primes 80–100 are \(\{83,89,97\}\); odd 90–100 are \(\{91,93,95,97,99\}\). So \(X\cup Y=\{83,89,91,93,95,97,99\}\); \(X\cap Y=\{97\}\).
(8) Quadrilaterals \(X\), rhombuses \(Y\), squares \(S\), rectangles \(V\), parallelograms \(T\). Write subset relations.
\(S\subseteq Y\), \(S\subseteq V\), \(Y\subseteq T\), \(V\subseteq T\), \(T\subseteq X\). Hence \(S\subseteq Y\subseteq T\subseteq X\) and \(S\subseteq V\subseteq T\subseteq X\).
(9) If \(M\) is any set, write \(M\cup \varnothing\) and \(M\cap \varnothing\).
\(M\cup \varnothing=M\); \(M\cap \varnothing=\varnothing\).
(10*) From the given Venn diagram (numbers placed in regions), write \(U,A,B,A\cup B,A\cap B\).
Pick all numbers in rectangle for \(U\); those inside circle \(A\) for \(A\); inside circle \(B\) for \(B\); union is all numbers in either circle; intersection is those in the overlap. (Exact lists depend on the labeled figure in your book.)
(11) If \(n(A)=7\), \(n(B)=13\), \(n(A\cap B)=4\), find \(n(A\cup B)\).
\(7+13-4=16\).
Activity I: With \(U=\{1,3,5,8,9,10,11,12,13,15\}\), \(A=\{1,11,13\}\), \(B=\{8,5,10,11,15\}\). Fill: \(A\cap B\), \(A\cup B\), \((A\cap B)'\), \(A'\), \(B'\), \(A'\cap B'\), \(A'\cup B'\), \((A\cup B)'\). Verify De Morgan laws.
\(A\cap B=\{11\}\). \(A\cup B=\{1,5,8,10,11,13,15\}\).
\(A'=U\setminus A=\{3,5,8,9,10,12,15\}\). \(B'=\{1,3,9,11,12,13\}\).
\((A\cap B)'=U\setminus\{11\}=\{1,3,5,8,9,10,12,13,15,11\}\) (i.e., all except 11).
\(A'\cap B'=\{3,9,12,13\}\). \(A'\cup B'=\{1,3,5,8,9,10,11,12,13,15\}\) (which is \(U\)).
\((A\cap B)'=A'\cup B'\) and \((A\cup B)'=A'\cap B'\) ✓.
\(A'=U\setminus A=\{3,5,8,9,10,12,15\}\). \(B'=\{1,3,9,11,12,13\}\).
\((A\cap B)'=U\setminus\{11\}=\{1,3,5,8,9,10,12,13,15,11\}\) (i.e., all except 11).
\(A'\cap B'=\{3,9,12,13\}\). \(A'\cup B'=\{1,3,5,8,9,10,11,12,13,15\}\) (which is \(U\)).
\((A\cap B)'=A'\cup B'\) and \((A\cup B)'=A'\cap B'\) ✓.
Mini Reference
Symbols
\(\in,\ \notin,\ \subseteq,\ =,\ \neq,\ \varnothing,\ U,\ A',\ \cup,\ \cap\)
Key Identities
\((A')'=A\), \(\ A\cup A'=U\), \(\ A\cap A'=\varnothing\)
\((A\cup B)'=A'\cap B'\), \((A\cap B)'=A'\cup B'\)
\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)
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