A set is a well-defined collection of distinct objects, called its elements or members. For example, the set of vowels in English = {a, e, i, o, u}.

What is the difference between a finite and an infinite set?

A finite set has a countable (limited) number of elements; an infinite set has an unlimited number of elements, such as the set of natural numbers.

A set A is a subset of a set B if every element of A is also an element of B, written A ⊆ B.

What are the union and intersection of sets?

The union A ∪ B is the set of all elements that are in A or B or both; the intersection A ∩ B is the set of elements common to both A and B.

What is the complement of a set?

The complement of a set A (written A′) is the set of all elements in the universal set that are not in A.

Samacheer Class 9 Maths - Set Language

Q: Represent the following sets in roster form.

(i) $A = {2,4,6,8,10,12,14,16,18}$

Q: Represent the following sets in set-builder form.

(i) $B = {x : x \text{ is an Indian cricket player who scored a double century in ODI}}$

Q: Represent the following sets in descriptive form.

(i) $P = {January, June, July}$

Q: Identify the following sets as finite or infinite.

(i) Set of all districts in Tamil Nadu → Finite (ii) Set of all straight lines passing through a point → Infinite (iii) $A={x:x\in\mathbb{Z},x<5}$

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Your Progress — Unit 1: Set Language0% complete

Ex 1.1Set6 questions

Q.1 Which of the following are sets? ▾

✓ Solution

(i) The collection of prime numbers up to 100.
✓ Set

(ii) The collection of rich people in India.
✕ Not a set

(iii) The collection of all rivers in India.
✓ Set

(iv) The collection of good Hockey players.
✕ Not a set

Q.2 List the set of letters of the following words in roster form. ▾

✓ Solution

(i) INDIA

$${I, N, D, A}$$

(ii) PARALLELOGRAM

$${P, A, R, L, E, O, G, M}$$

(iii) MISSISSIPPI

$${M, I, S, P}$$

(iv) CZECHOSLOVAKIA

$${C, Z, E, H, O, S, L, V, A, K, I}$$

Q.3 Consider the following sets ▾

✓ Solution

$$A = {0, 3, 5, 8}$$

$$B = {2, 4, 6, 10}$$

$$C = {12, 14, 18, 20}$$

(a) State whether True or False

(i) (18 \in C) → True

(ii) (6 \notin A) → True

(iii) (14 \notin C) → False

(iv) (10 \in B) → True

(v) (5 \in B) → False

(vi) (0 \in B) → False

(b) Fill in the blanks

(i) (3 \in \boxed{A})

(ii) (14 \in \boxed{C})

(iii) (18 \boxed{\notin} B)

(iv) (4 \boxed{\in} B)

Q.4 Represent the following sets in roster form. ▾

✓ Solution

(i)

$$A = {2,4,6,8,10,12,14,16,18}$$

(ii)

B = \left{\frac12,\frac22,\frac32,\frac42,\frac52\right}

B = \left{\frac12,1,\frac32,2,\frac52\right}

(iii)

Perfect cubes between 27 and 216 are:

64,\ 125

Hence,

$$C = {64,125}$$

(iv)

$$D = {-4,-3,-2,-1,0,1,2}$$

Q.5 Represent the following sets in set-builder form. ▾

✓ Solution

(i)

B = {x : x \text{ is an Indian cricket player who scored a double century in ODI}}

(ii)

C = \left{x : x=\frac{n}{n+1},\ n\in\mathbb{N}\right}

(iii)

D = {x : x \text{ is a Tamil month in a year}}

(iv)

E = {x : x \text{ is an odd whole number and } x<9}

Q.6 Represent the following sets in descriptive form. ▾

✓ Solution

(i)

$$P = {January, June, July}$$

Descriptive Form:
The set of months whose names begin with the letter “J”.

(ii)

$$Q = {7,11,13,17,19,23,29}$$

Descriptive Form:
The set of prime numbers between 5 and 30.

(iii)

R = {x : x \in \mathbb{N}, x<5}

Descriptive Form:
The set of natural numbers less than 5.

(iv)

S = {x : x \text{ is a consonant in English alphabets}}

Descriptive Form:
The set of consonants in the English alphabet.

Ex 1.2Types of Sets10 questions

Q.1 Find the cardinal number of the following sets. ▾

✓ Solution

(i)

$$M = {p,q,r,s,t,u}$$

$$n(M)=6$$

(ii)

P = {x : x=3n+2,\ n\in W,\ x<15}

Values:

$$2,5,8,11,14$$

$$n(P)=5$$

(iii)

Q = \left{y : y=\frac{4}{3n},\ n\in\mathbb{N},\ 2<n\le5\right}

Values:

\left{\frac49,\frac13,\frac4{15}\right}

$$n(Q)=3$$

(iv)

R={x:x\in\mathbb{Z}, -5\le x<5}

Elements:

$${-5,-4,-3,-2,-1,0,1,2,3,4}$$

$$n(R)=10$$

(v)

Leap years between 1882 and 1906:

$$1884,1888,1892,1896,1904$$

$$n(S)=5$$

Q.2 Identify the following sets as finite or infinite. ▾

✓ Solution

(i) Set of all districts in Tamil Nadu → Finite

(ii) Set of all straight lines passing through a point → Infinite

(iii)

A={x:x\in\mathbb{Z},x<5}

→ Infinite

(iv)

B={x:x^2-5x+6=0,\ x\in\mathbb{N}}

Roots are (2,3)

→ Finite

Q.3 Which of the following sets are equivalent, unequal, or equal? ▾

✓ Solution

(i)

A = vowels in English alphabet

$$A={a,e,i,o,u}$$

B = letters in “VOWEL”

$$B={V,O,W,E,L}$$

Both have 5 elements.

✓ Equivalent sets

(ii)

$$C={2,3,4,5}$$

D={x:x\in W,\ 1<x<5}={2,3,4}

✕ Unequal sets

(iii)

$$X={L,I,F,E}$$

$$Y={F,I,L,E}$$

✓ Equal sets

(iv)

$$G={5,7,11,13,17,19}$$

$$H={1,2,3,6,9,18}$$

Both contain 6 elements.

✓ Equivalent sets

Q.4 Identify the following sets as null set or singleton set. ▾

✓ Solution

(i)

A={x:x\in\mathbb{N},1<x<2}

No natural number exists.

✓ Null set

(ii)

Set of even natural numbers not divisible by 2.

Impossible.

✓ Null set

(iii)

$$C={0}$$

✓ Singleton set

(iv)

Set of triangles having four sides.

Impossible.

✓ Null set

Q.5 State which pairs of sets are disjoint or overlapping. ▾

✓ Solution

(i)

$$A={f,i,a,s}$$

$$B={a,n,f,h,s}$$

Common elements:

$${a,f,s}$$

✓ Overlapping sets

(ii)

C = odd prime numbers greater than 2

D = even prime number

No common element.

✓ Disjoint sets

(iii)

Factors of 24:

$$E={1,2,3,4,6,8,12,24}$$

Multiples of 3 less than 30:

$$F={3,6,9,12,15,18,21,24,27}$$

Common elements:

$${3,6,12,24}$$

✓ Overlapping sets

Q.6 If S = {square, rectangle, circle, rhombus, triangle}, find: (i) Shapes having 4 equal sides; (ii) Shapes having radius; (iii) Shapes whose interior angles sum to 180°; (iv) Shapes having 5 sides. ▾

(i) square, rhombus

(ii) circle

(iii) triangle

(iv) ∅

Q.7 If A = {a, {a,b}}, write all subsets (the power set P(A)). ▾

P(A) = {∅, {a}, {{a,b}}, {a, {a,b}}}

Q.8 Write the power set. ▾

✓ Solution

(i)

$$A={a,b}$$

P(A)={\varnothing,{a},{b},{a,b}}

(ii)

$$B={1,2,3}$$

P(B)= { \varnothing, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }

(iii)

$$D={p,q,r,s}$$

Number of subsets:

$$2^4=16$$

(iv)

E=\varnothing

P(E)={\varnothing}

Q.9 Find the number of subsets and proper subsets. ▾

✓ Solution

(i)

W={\text{red, blue, yellow}}

$$n(W)=3$$

Number of subsets:

$$2^3=8$$

Proper subsets:

$$8-1=7$$

(ii)

X={x^2:x\in\mathbb{N},x^2\le100}

Squares:

$${1,4,9,16,25,36,49,64,81,100}$$

$$n(X)=10$$

Subsets:

2^{10}=1024

Proper subsets:

$$1023$$

Q.10 Question 10 ▾

✓ Solution

(i)

If (n(A)=4)

$$n[P(A)] = 2^4 = 16$$

(ii)

If (n(A)=0)

$$n[P(A)] = 2^0 = 1$$

(iii)

$$n[P(A)] =256$$

Then,

$$2^n=256$$

$$n=8$$

$$n(A)=8$$

Ex 1.3Set Operations7 questions

Q.1 Using the given Venn diagram, write the elements of: ▾

✓ Solution

The Venn diagram is not provided in the question.
So, the elements of the sets cannot be determined exactly.

Generally:

(i) (A) → all elements inside set (A)
(ii) (B) → all elements inside set (B)
(iii) (A \cup B) → elements in (A) or (B) or both
(iv) (A \cap B) → common elements of (A) and (B)
(v) (A - B) → elements in (A) but not in (B)
(vi) (B - A) → elements in (B) but not in (A)
(vii) (A') → elements not in (A)
(viii) (B') → elements not in (B)
(ix) (U) → universal set

Q.2 Find (A \cup B), (A \cap B), (A-B) and (B-A) ▾

✓ Solution

(i) (A={2,6,10,14}), (B={2,5,14,16})

Solution

A \cup B = {2,5,6,10,14,16}

A \cap B = {2,14}

$$A-B = {6,10}$$

$$B-A = {5,16}$$

(ii) (A={a,b,c,e,u}), (B={a,e,i,o,u})

Solution

A \cup B = {a,b,c,e,i,o,u}

A \cap B = {a,e,u}

$$A-B = {b,c}$$

$$B-A = {i,o}$$

(iii)

A={x:x\in N,\ x\le10}

B={x:x\in W,\ x<6}

Step 1: Write the sets

$$A={1,2,3,4,5,6,7,8,9,10}$$

$$B={0,1,2,3,4,5}$$

Solution

A\cup B={0,1,2,3,4,5,6,7,8,9,10}

A\cap B={1,2,3,4,5}

$$A-B={6,7,8,9,10}$$

$$B-A={0}$$

(iv)

(A) = set of letters in “mathematics”

(B) = set of letters in “geometry”

Step 1: Write the sets

$$A={m,a,t,h,e,i,c,s}$$

$$B={g,e,o,m,t,r,y}$$

Solution

A\cup B={a,c,e,g,h,i,m,o,r,s,t,y}

A\cap B={m,e,t}

$$A-B={a,h,i,c,s}$$

$$B-A={g,o,r,y}$$

Q.3 Let U = {a,b,c,d,e,f,g,h}, A = {b,d,f,h} and B = {a,d,e,h}. Find: (i) A' (ii) B' (iii) A' ∪ B' (iv) A' ∩ B' (v) (A ∪ B)' (vi) (A ∩ B)' (vii) (A')' (viii) (B')'. ▾

A' = {a, c, e, g}
B' = {b, c, f, g}
A' ∪ B' = {a, b, c, e, f, g}
A' ∩ B' = {c, g}
(A ∪ B)' = {c, g}
(A ∩ B)' = {a, b, c, e, f, g}
(A')' = A = {b, d, f, h}
(B')' = B = {a, d, e, h}

Q.4 Let U = {0,1,2,3,4,5,6,7}, A = {1,3,5,7} and B = {0,2,3,5,7}. Find: (i) A' (ii) B' (iii) A' ∪ B' (iv) A' ∩ B' (v) (A ∪ B)' (vi) (A ∩ B)' (vii) (A')' (viii) (B')'. ▾

A' = {0,2,4,6}
B' = {1,4,6}
A' ∪ B' = {0,1,2,4,6}
A' ∩ B' = {4,6}
(A ∪ B)' = {4,6}
(A ∩ B)' = {0,1,2,4,6}
(A')' = A = {1,3,5,7}
(B')' = B = {0,2,3,5,7}

Q.5 Find the symmetric difference between the sets ▾

✓ Solution

The symmetric difference is:

A \triangle B = (A-B)\cup(B-A)

(i)

$$P={2,3,5,7,11}$$

$$Q={1,3,5,11}$$

Solution

$$P-Q={2,7}$$

$$Q-P={1}$$

P\triangle Q={1,2,7}

(ii)

$$R={l,m,n,o,p}$$

$$S={j,l,n,q}$$

Solution

$$R-S={m,o,p}$$

$$S-R={j,q}$$

R\triangle S={j,m,o,p,q}

(iii)

$$X={5,6,7}$$

$$Y={5,7,9,10}$$

Solution

$$X-Y={6}$$

$$Y-X={9,10}$$

X\triangle Y={6,9,10}

Q.6 Using the set symbols, write down the expressions for the shaded region ▾

✓ Solution

The shaded diagrams are not provided in the question.
Hence the exact expressions cannot be determined.

Common shaded regions are:

| Region | Expression |
| -------------- | ------------ |
| Common part | (A \cap B) |
| Entire portion | (A \cup B) |
| Only A | (A-B) |
| Only B | (B-A) |
| Outside both | ((A\cup B)') |

Q.7 Draw Venn diagrams for the following: (i) A ∪ B (ii) A ∩ B (iii) (A ∩ B)' (iv) (B − A)' (v) A' ∪ B' (vi) A' ∩ B' (vii) State any observation. ▾

Venn shading instructions

(i) A ∪ B — Shade both sets A and B (all regions inside A or B).
(ii) A ∩ B — Shade only the common overlapping region of A and B.
(iii) (A ∩ B)' — Shade every region except the intersection (i.e., all regions not in A ∩ B).
(iv) (B − A)' — Shade everything except the part belonging only to B (i.e., the complement of B\A).
(v) A' ∪ B' — Shade all regions except the intersection (equivalent to (A ∩ B)').
(vi) A' ∩ B' — Shade the region outside both sets (elements in neither A nor B).
(vii) Observation — (A ∩ B)' = A' ∪ B' (De Morgan's law).

Ex 1.4Properties of Set Operations14 questions

Important Properties

Q.1 Commutative Laws ▾

✓ Solution

A\cup B = B\cup A

A\cap B = B\cap A

Q.2 Associative Laws ▾

✓ Solution

(A\cup B)\cup C = A\cup(B\cup C)

(A\cap B)\cap C = A\cap(B\cap C)

Q.3 Distributive Laws ▾

✓ Solution

A\cup(B\cap C) = (A\cup B)\cap(A\cup C)

A\cap(B\cup C) = (A\cap B)\cup(A\cap C)

Q.4 Identity Laws ▾

✓ Solution

A\cup \varnothing = A

A\cap U = A

Q.5 Domination Laws ▾

✓ Solution

A\cup U = U

A\cap \varnothing = \varnothing

Q.6 Idempotent Laws ▾

✓ Solution

A\cup A = A

A\cap A = A

Q.7 Complement Laws ▾

✓ Solution

A\cup A' = U

A\cap A' = \varnothing

Q.8 Double Complement Law ▾

✓ Solution

$$(A')' = A$$

Q.9 De Morgan’s Laws ▾

✓ Solution

(A\cup B)' = A'\cap B'

(A\cap B)' = A'\cup B'

Q.1 If P = {1,2,5,7,9}, Q = {2,3,5,9,11}, R = {3,4,5,7,9}, S = {2,3,4,5,8}, find: (i) (P ∪ Q) ∪ R; (ii) (P ∩ Q) ∩ S; (iii) (Q ∩ S) ∩ R. ▾

(i) (P ∪ Q) = {1,2,3,5,7,9,11}; (P ∪ Q) ∪ R = {1,2,3,4,5,7,9,11}

(ii) P ∩ Q = {2,5,9}; (P ∩ Q) ∩ S = {2,5}

(iii) Q ∩ S = {2,3,5}; (Q ∩ S) ∩ R = {3,5}

Q.2 Test the commutative property of union and intersection ▾

✓ Solution

Given:

P={x:x\text{ is a real number between }2\text{ and }7}

Q={x:x\text{ is a rational number between }2\text{ and }7}

Since every rational number is a real number,

Q\subset P

Union

P\cup Q=P

Also,

Q\cup P=P

Therefore,

P\cup Q=Q\cup P

Hence, the commutative property of union is verified.

Intersection

P\cap Q=Q

Also,

Q\cap P=Q

Therefore,

P\cap Q=Q\cap P

Hence, the commutative property of intersection is verified.

Q.3 Verify the associative property of union ▾

✓ Solution

Given:

$$A={p,q,r,s}$$

$$B={m,n,q,s,t}$$

$$C={m,n,p,q,s}$$

We need to verify:

(A\cup B)\cup C=A\cup(B\cup C)

Left Side

Step 1: Find (A\cup B)

A\cup B={m,n,p,q,r,s,t}

Step 2: Find ((A\cup B)\cup C)

{m,n,p,q,r,s,t}\cup{m,n,p,q,s}

$$={m,n,p,q,r,s,t}$$

Right Side

Step 1: Find (B\cup C)

B\cup C={m,n,p,q,s,t}

Step 2: Find (A\cup(B\cup C))

{p,q,r,s}\cup{m,n,p,q,s,t}

$$={m,n,p,q,r,s,t}$$

Conclusion

(A\cup B)\cup C=A\cup(B\cup C)

Hence, the associative property of union is verified.

Q.4 Verify the associative property of intersection ▾

✓ Solution

Given:

A={-11,\sqrt2,\sqrt5,7}

B={\sqrt3,\sqrt5,6,13}

C={\sqrt2,\sqrt3,\sqrt5,9}

We need to verify:

(A\cap B)\cap C=A\cap(B\cap C)

Left Side

Step 1: Find (A\cap B)

Common element:

A\cap B={\sqrt5}

Step 2: Find ((A\cap B)\cap C)

{\sqrt5}\cap{\sqrt2,\sqrt3,\sqrt5,9}

={\sqrt5}

Right Side

Step 1: Find (B\cap C)

B\cap C={\sqrt3,\sqrt5}

Step 2: Find (A\cap(B\cap C))

{-11,\sqrt2,\sqrt5,7}\cap{\sqrt3,\sqrt5}

={\sqrt5}

Conclusion

(A\cap B)\cap C=A\cap(B\cap C)

Hence, the associative property of intersection is verified.

Q.5 Verify the associative property of intersection of sets ▾

✓ Solution

Given:

A={x:x=2n,\ n\in W,\ n<4}

B={x:x=2n,\ n\in N,\ n\le4}

$$C={0,1,2,5,6}$$

Step 1: Write the sets explicitly

Set (A)

Since (n\in W) and (n<4),

$$n=0,1,2,3$$

Therefore,

$$A={0,2,4,6}$$

Set (B)

Since (n\in N) and (n\le4),

$$n=1,2,3,4$$

Therefore,

$$B={2,4,6,8}$$

Set (C)

$$C={0,1,2,5,6}$$

We verify:

(A\cap B)\cap C=A\cap(B\cap C)

Left Side

Step 1: Find (A\cap B)

A\cap B={2,4,6}

Step 2: Find ((A\cap B)\cap C)

{2,4,6}\cap{0,1,2,5,6}

$$={2,6}$$

Right Side

Step 1: Find (B\cap C)

B\cap C={2,6}

Step 2: Find (A\cap(B\cap C))

{0,2,4,6}\cap{2,6}

$$={2,6}$$

Conclusion

(A\cap B)\cap C=A\cap(B\cap C)

Hence, the associative property of intersection is verified.

Ex 1.5De Morgan’s Laws0 questions

De Morgan’s Laws

For any two sets (A) and (B),

First Law

(A\cup B)' = A'\cap B'

Meaning:

> The complement of union equals the intersection of complements.

Second Law

(A\cap B)' = A'\cup B'

Meaning:

> The complement of intersection equals the union of complements.

# Important Results

Complement Laws

A\cup A'=U

A\cap A'=\varnothing

Double Complement Law

$$(A')'=A$$

Identity Laws

A\cup\varnothing=A

A\cap U=A

Domination Laws

A\cup U=U

A\cap\varnothing=\varnothing

Ex 1.6Application on Cardinality of Sets11 questions

# Important Formula

For any two sets (A) and (B):

genui{"math_block_widget_always_prefetch_v2":{"content":"n(A\cup B)=n(A)+n(B)-n(A\cap B)"}}

For three sets:

n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)

Q.1 Given: n(A) = 25, n(B) = 40, n(A ∪ B) = 50 and n(B') = 25. Find (1) n(A ∩ B) and (2) n(U). ▾

✓ Solution

Given:

$$n(A)=25$$

$$n(B)=40$$

n(A\cup B)=50

$$n(B')=25$$

Find:

1. (n(A\cap B))
2. (n(U))

Step 1: Find (n(A\cap B))

Using:

n(A\cup B)=n(A)+n(B)-n(A\cap B)

50=25+40-n(A\cap B)

50=65-n(A\cap B)

n(A\cap B)=15

Step 2: Find (n(U))

Since:

$$n(B')=n(U)-n(B)$$

$$25=n(U)-40$$

$$n(U)=65$$

Answer

n(A\cap B)=15

$$n(U)=65$$

Q.1 Given: n(A) = 300, n(A ∪ B) = 500, n(A ∩ B) = 50 and n(B') = 350. Find (1) n(B) and (2) n(U). ▾

✓ Solution

Given:

$$n(A)=300$$

n(A\cup B)=500

n(A\cap B)=50

$$n(B')=350$$

Find:

1. (n(B))
2. (n(U))

Step 1: Find (n(B))

n(A\cup B)=n(A)+n(B)-n(A\cap B)

$$500=300+n(B)-50$$

$$500=250+n(B)$$

$$n(B)=250$$

Step 2: Find (n(U))

$$n(B')=n(U)-n(B)$$

$$350=n(U)-250$$

$$n(U)=600$$

Answer

$$n(B)=250$$

$$n(U)=600$$

Q.2 Verify n(A ∪ B) = n(A) + n(B) − n(A ∩ B) for U = {x : x ∈ N, x ≤ 10}, A = {2,3,4,8,10} and B = {1,2,5,8,10}. ▾

✓ Solution

n(A\cup B)=n(A)+n(B)-n(A\cap B)

Given:

U={x:x\in N,\ x\le10}

$$A={2,3,4,8,10}$$

$$B={1,2,5,8,10}$$

Step 1: Find (n(A))

$$n(A)=5$$

Step 2: Find (n(B))

$$n(B)=5$$

Step 3: Find (A\cap B)

A\cap B={2,8,10}

n(A\cap B)=3

Step 4: Find (A\cup B)

A\cup B={1,2,3,4,5,8,10}

n(A\cup B)=7

Verification

RHS:

n(A)+n(B)-n(A\cap B)

$$=5+5-3$$

$$=7$$

LHS:

n(A\cup B)=7

Thus,

n(A\cup B)=n(A)+n(B)-n(A\cap B)

Verified.

Q.3 Verify the formula n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C) for A = {a,c,e,f,h}, B = {c,d,e,f} and C = {a,b,c,f}. ▾

✓ Solution

n(A\cup B\cup C)

$$=n(A)+n(B)+n(C)$$

-n(A\cap B)-n(B\cap C)-n(A\cap C)

+n(A\cap B\cap C)

Q.3 Given A = {a, c, e, f, h}, B = {c, d, e, f}, C = {a, b, c, f}. Verify that n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). ▾

Given sets:

n(A) = 5
n(B) = 4
n(C) = 4

Intersections:

A ∩ B = {c, e, f} ⇒ n(A ∩ B) = 3
B ∩ C = {c, f} ⇒ n(B ∩ C) = 2
A ∩ C = {a, c, f} ⇒ n(A ∩ C) = 3
A ∩ B ∩ C = {c, f} ⇒ n(A ∩ B ∩ C) = 2

Compute RHS:

n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)

= 5 + 4 + 4 − 3 − 2 − 3 + 2 = 7

Union:

A ∪ B ∪ C = {a, b, c, d, e, f, h} ⇒ n(A ∪ B ∪ C) = 7

Hence both sides equal 7; the formula is verified.

Q.3 Given A = {1, 3, 5}, B = {2, 3, 5, 6}, C = {1, 5, 6, 7}. Verify that n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). ▾

Given:

n(A) = 3
n(B) = 4
n(C) = 4

Intersections:

A ∩ B = {3, 5} ⇒ n(A ∩ B) = 2
B ∩ C = {5, 6} ⇒ n(B ∩ C) = 2
A ∩ C = {1, 5} ⇒ n(A ∩ C) = 2
A ∩ B ∩ C = {5} ⇒ n(A ∩ B ∩ C) = 1

Compute RHS:

3 + 4 + 4 − 2 − 2 − 2 + 1 = 6

Union:

A ∪ B ∪ C = {1, 2, 3, 5, 6, 7} ⇒ n(A ∪ B ∪ C) = 6

Therefore the formula is verified.

Q.4 Music and Drama Problem ▾

✓ Solution

Given:

Students in music = 25
Students in drama = 30
Students in both = 8

(i) Only music

$$25-8=17$$

Answer

$$17$$

(ii) Only drama

$$30-8=22$$

Answer

$$22$$

(iii) Total students

n(M\cup D)=n(M)+n(D)-n(M\cap D)

$$=25+30-8$$

$$=47$$

Answer

$$47$$

Q.5 Tea and Coffee Problem ▾

✓ Solution

Given:

Total people = 45
Tea = 35
Coffee = 20

Everyone likes tea or coffee or both.

(i) Like both tea and coffee

45=35+20-n(T\cap C)

45=55-n(T\cap C)

n(T\cap C)=10

Answer

$$10$$

(ii) Do not like tea

$$45-35=10$$

Answer

$$10$$

(iii) Do not like coffee

$$45-20=25$$

Answer

$$25$$

Q.6 Examination Problem ▾

✓ Solution

Given:

50% passed Mathematics
70% passed Science
10% failed both
300 passed both

Find total students.

Step 1: Passed at least one subject

Since 10% failed both,

$$90%$$

passed at least one subject.

Step 2: Let total students = (x)

Mathematics:

50% \text{ of }x=0.5x

Science:

70% \text{ of }x=0.7x

Both:

$$300$$

Using formula:

$$0.9x=0.5x+0.7x-300$$

$$0.9x=1.2x-300$$

$$300=0.3x$$

$$x=1000$$

Answer

$$1000$$

students appeared.

Q.7 Venn Diagram Problem ▾

✓ Solution

Given:

$$n(A-B)=32+x$$

$$n(B-A)=5x$$

n(A\cap B)=x

Also,

$$n(A)=n(B)$$

Step 1: Find (n(A))

n(A)=n(A-B)+n(A\cap B)

$$=(32+x)+x$$

$$=32+2x$$

Step 2: Find (n(B))

n(B)=n(B-A)+n(A\cap B)

$$=5x+x$$

$$=6x$$

Step 3: Since (n(A)=n(B))

$$32+2x=6x$$

$$32=4x$$

$$x=8$$

Answer

$$x=8$$

Q.8 Car Owners Problem ▾

✓ Solution

Given:

Total investigated = 500
Car A owners = 400
Car B owners = 200
Both A and B = 50

Using formula:

n(A\cup B)=n(A)+n(B)-n(A\cap B)

$$=400+200-50$$

$$=550$$

But total people investigated = 500.

This is impossible because total owners cannot exceed total people.

Answer

The data is not correct.

Brain Grain Grade 9 Maths

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Unit 2: Real Numbers

Samacheer Class 9 Maths - Set Language

Set Language — key concepts & quick answers

(i) INDIA

(ii) PARALLELOGRAM

(iii) MISSISSIPPI

(iv) CZECHOSLOVAKIA

(a) State whether True or False

(b) Fill in the blanks

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(v)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(i)

(ii)

(iii)

(i) (A={2,6,10,14}), (B={2,5,14,16})

Solution

(ii) (A={a,b,c,e,u}), (B={a,e,i,o,u})

Solution

(iii)

Step 1: Write the sets

Solution

(iv)

Step 1: Write the sets

Solution

(i)

Solution

(ii)

Solution

(iii)

Solution

Important Properties

Union

Intersection

Left Side

Step 1: Find (A\cup B)

Step 2: Find ((A\cup B)\cup C)

Right Side

Step 1: Find (B\cup C)

Step 2: Find (A\cup(B\cup C))

Conclusion

Left Side

Step 1: Find (A\cap B)

Step 2: Find ((A\cap B)\cap C)

Right Side

Step 1: Find (B\cap C)

Step 2: Find (A\cap(B\cap C))

Conclusion

Step 1: Write the sets explicitly

Set (A)