SETS | Jamb Mathematics

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"Get ready to explore the concept of 'Set' and discover how it applies to everyday life! Understanding sets helps us organize, categorize, and make sense of the world around us, from grouping objects to making informed decisions. As we dive into this topic, think about how sets shape the way we interact with people, ideas, and experiences in daily life."

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Sets? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Sets together and move one step closer to achieving your exam success! Blissful learning.

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calculation-based questions along with their answers covering different types of sets:

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1. Empty Set (Null Set)

Q1: If

A = \{ x \in \mathbb{Z} \mid x^2 = -1 \}

, what is

n(A)

, the number of elements in

A

A: Since no integer satisfies $x^2 = -1$ , $A$ is an empty set. Thus, $n(A) = 0$ .

Q2: Let

B = \{ x \mid x

is a prime number less than 2 }. Determine

|B|

A: The only prime number less than 2 is 1, which is not prime. So, $B$ is empty, and $|B| = 0$ .

2. Universal Set

Q3: If the universal set is

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

, and

A = \{2,4,6,8\}

, find

|U - A|

A: The complement of $A$ in $U$ is $U - A = \{1,3,5,7,9,10\}$ , so $|U - A| = 6$ .

Q4: Given

U = \{x \mid 1 \leq x \leq 20, x \in \mathbb{N} \}

and

A = \{ x \mid x

is an even number in

U

}, find

|A^c|

A: The even numbers in $U$ are $\{2,4,6,8,10,12,14,16,18,20\}$ , so $|A| = 10$ . The complement $A^c$ contains the odd numbers, also 10 in count. Thus, $|A^c| = 10$ .

3. Complement of a Set

Q5: If

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

and

A = \{2,4,6,8,10\}

, compute

|A^c|

A: The complement $A^c = U - A = \{1,3,5,7,9\}$ , so $|A^c| = 5$ .

Q6: Given

U = \{ x \in \mathbb{Z} \mid -5 \leq x \leq 5 \}

and

A = \{ x \mid x

is a non-negative integer}, find

|A^c|

A: The non-negative integers are $\{0,1,2,3,4,5\}$ , so $|A| = 6$ . The complement $A^c = \{-5,-4,-3,-2,-1\}$ , so $|A^c| = 5$ .

4. Subsets

Q7: How many subsets does the set

A = \{1,2,3,4\}

have?

A: The number of subsets is $2^n = 2^4 = 16$ .

Q8: If

B = \{a,b,c,d,e\}

, find the number of proper subsets of

B

A: The total subsets are $2^5 = 32$ , and the number of proper subsets is $32 - 1 = 31$ .

5. Finite and Infinite Sets

Q9: If

C = \{x \mid x

is a multiple of 3 and

x < 30

}, find

|C|

A: The elements are $\{3,6,9,12,15,18,21,24,27\}$ , so $|C| = 9$ .

Q10: If

D = \{ x \mid x

is a perfect square and

x \leq 100 \}

, find

|D|

A: The perfect squares are $\{1,4,9,16,25,36,49,64,81,100\}$ , so $|D| = 10$ .

6. Infinite Set

Q11: Let

E = \{ x \mid x

is a prime number}. Is

E

finite or infinite?

A: The set of prime numbers is infinite.

Q12: The set

F = \{ x \mid x

is a natural number greater than 0 }. Determine if

F

is finite or infinite.

A: The set of natural numbers is infinite.

7. Disjoint Sets

Q13: If

G = \{1,2,3,4,5\}

and

H = \{6,7,8,9,10\}

, find

|G \cap H|

A: The sets have no common elements, so $|G \cap H| = 0$ .

Q14: Given

I = \{x \mid x

is an even number} and

J = \{x \mid x

is an odd number}, are

I

and

J

disjoint?

A: Yes, because no number can be both even and odd.

8. Union and Intersection of Sets

Q15: If

K = \{1,3,5,7\}

and

L = \{2,4,6,8\}

, find

|K \cup L|

A: $|K \cup L| = |K| + |L| = 4 + 4 = 8$ .

Q16: Given

M = \{a,b,c,d\}

and

N = \{b,c,e,f\}

, find

|M \cap N|

A: The common elements are $\{b,c\}$ , so $|M \cap N| = 2$ .

9. Cartesian Product

Q17: If

A = \{1,2\}

and

B = \{x,y,z\}

, find

|A \times B|

A: The Cartesian product has $|A| \times |B| = 2 \times 3 = 6$ elements.

Q18: If

P = \{a,b,c\}

and

Q = \{1,2,3,4\}

, how many ordered pairs exist in

P \times Q

A: $|P \times Q| = 3 \times 4 = 12$ .

10. Power Set

Q19: If

S = \{a,b\}

, how many elements are in the power set

P(S)

A: The power set contains $2^2 = 4$ elements: $\{\emptyset, \{a\}, \{b\}, \{a,b\} \}$ .

Q20: If

T = \{1,2,3,4,5\}

, find

|P(T)|

A: The number of subsets is $2^5 = 32$ , so $|P(T)| = 32$ .
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Calculation problems involving Cardinality of Sets

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1. Basic Cardinality

Q1: If

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

, find

|A|

A: The set has 10 elements, so $|A| = 10$ .

Q2: Given

B = \{a, b, c, d, e\}

, determine

|B|

A: $B$ contains 5 elements, so $|B| = 5$ .

2. Union of Two Sets

Q3: If

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

and

B = \{4,5,6,7,8\}

, find

|A \cup B|

A: $A \cup B = \{1,2,3,4,5,6,7,8\}$ , so $|A \cup B| = 8$ .

Q4: If

X = \{a, b, c\}

and

Y = \{b, c, d, e\}

, compute

|X \cup Y|

A: $X \cup Y = \{a, b, c, d, e\}$ , so $|X \cup Y| = 5$ .

3. Intersection of Two Sets

Q5: If

A = \{2,4,6,8,10\}

and

B = \{1,2,3,4,5,6\}

, find

|A \cap B|

A: $A \cap B = \{2,4,6\}$ , so $|A \cap B| = 3$ .

Q6: If

M = \{x,y,z\}

and

N = \{w,x,y,z\}

, compute

|M \cap N|

A: $M \cap N = \{x,y,z\}$ , so $|M \cap N| = 3$ .

4. Cardinality Using the Inclusion-Exclusion Principle

Q7: Given

|P| = 12

|Q| = 18

, and

|P \cap Q| = 5

, find

|P \cup Q|

A: Using $|P \cup Q| = |P| + |Q| - |P \cap Q|$ , we get $12 + 18 - 5 = 25$ .

Q8: If

|A| = 20

|B| = 15

, and

|A \cap B| = 7

, find

|A \cup B|

A: $|A \cup B| = 20 + 15 - 7 = 28$ .

5. Complement of a Set

Q9: If the universal set

U

has 50 elements and

|A| = 30

, find

|A^c|

A: $|A^c| = |U| - |A| = 50 - 30 = 20$ .

Q10: Given

U = \{1,2,3,\dots,100\}

and

B = \{ x \mid x

is a multiple of 5 in

U

}, find

|B^c|

A: The multiples of 5 in $U$ are $\{5,10,15,\dots,100\}$ , which is 20 elements. $|B^c| = 100 - 20 = 80$ .

6. Disjoint Sets

Q11: If

C = \{1,3,5,7\}

and & D = {2,4,6,8}

, find

|C \cap D| $.

A: $C \cap D = \emptyset$ , so $|C \cap D| = 0$ .

Q12: Given

|X| = 40

|Y| = 25

, and

|X \cap Y| = 0

, determine

|X \cup Y|

A: Since $X$ and $Y$ are disjoint, $|X \cup Y| = |X| + |Y| = 40 + 25 = 65$ .

7. Subsets and Power Set

Q13: If

E = \{a, b, c, d\}

, find the number of subsets of

E

A: The number of subsets is $2^n = 2^4 = 16$ .

Q14: How many proper subsets does

F = \{1,2,3,4,5,6\}

have?

A: The total subsets are $2^6 = 64$ , and proper subsets are $64 - 1 = 63$ .

8. Cartesian Product

Q15: If

A = \{1,2,3\}

and

B = \{x,y\}

, find

|A \times B|

A: The number of ordered pairs is $|A| \times |B| = 3 \times 2 = 6$ .

Q16: If

P = \{a,b,c,d\}

and

Q = \{1,2,3,4,5\}

, compute

|P \times Q|

A: $|P \times Q| = 4 \times 5 = 20$ .

9. Difference of Sets

Q17: If

G = \{1,2,3,4,5,6\}

and

H = \{2,4,6\}

, compute

|G - H|

A: $G - H = \{1,3,5\}$ , so $|G - H| = 3$ .

Q18: Given

X = \{a,b,c,d,e\}

and

Y = \{b,d\}

, find

|X - Y|

A: $X - Y = \{a,c,e\}$ , so $|X - Y| = 3$ .

10. Partition of a Set

Q19: A set

S

is partitioned into three disjoint subsets

A, B, C

with

|A| = 10

|B| = 15

, and

|C| = 25

. Find

|S|

A: $|S| = |A| + |B| + |C| = 10 + 15 + 25 = 50$ .

Q20: If a set

T

is divided into four mutually exclusive subsets of sizes 8, 12, 20, and 30, find

|T|

A: $|T| = 8 + 12 + 20 + 30 = 70$ .

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Calculation problems involving not more than three (3) Sets

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1. Basic Cardinality of Sets (Single Set)

Q1: If a universal set

U

has 50 elements and a subset

A

has 20 elements, how many elements are in

A^c

A: $|A^c| = |U| - |A| = 50 - 20 = 30$ .

Q2: Given

|A| = 15

and

|B| = 25

, if

|A \cap B| = 5

, find

|A \cup B|

A: $|A \cup B| = |A| + |B| - |A \cap B| = 15 + 25 - 5 = 35$ .

2. Two-Set Problems (Union and Intersection)

Q3: In a group of 40 students, 18 study Mathematics, 25 study Science, and 10 study both subjects. How many study at least one subject?

A: $|M \cup S| = |M| + |S| - |M \cap S| = 18 + 25 - 10 = 33$ .

Q4: If a survey shows 30 people like coffee, 20 like tea, and 10 like both, how many like either coffee or tea?

A: $|C \cup T| = |C| + |T| - |C \cap T| = 30 + 20 - 10 = 40$ .

Q5: If

|A| = 100

|B| = 80

, and

|A \cup B| = 150

, find

|A \cap B|

A: $|A \cap B| = |A| + |B| - |A \cup B| = 100 + 80 - 150 = 30$ .

3. Complement and Only-One Membership

Q6: In a group of 60 people, 35 own a car, 25 own a bike, and 15 own both. How many own only a car?

A: $|C - B| = |C| - |C \cap B| = 35 - 15 = 20$ .

Q7: If 50 students are surveyed and 30 like basketball, 20 like soccer, and 10 like both, how many like only one sport?

A: $|B - S| + |S - B| = (30 - 10) + (20 - 10) = 20 + 10 = 30$ .

4. Three-Set Problems (Union and Intersection)

Q8: If

|A| = 50

|B| = 40

|C| = 30

|A \cap B| = 20

|B \cap C| = 10

|C \cap A| = 15

, and

|A \cap B \cap C| = 5

, find

|A \cup B \cup C|

A:
$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$ $= 50 + 40 + 30 - 20 - 10 - 15 + 5 = 80$

Q9: In a class of 80 students, 50 study Physics, 30 study Chemistry, and 40 study Biology. If 20 study both Physics and Chemistry, 15 study both Chemistry and Biology, 10 study both Physics and Biology, and 5 study all three, find how many study at least one subject.

A:
$|P \cup C \cup B| = |P| + |C| + |B| - |P \cap C| - |C \cap B| - |P \cap B| + |P \cap C \cap B|$ $= 50 + 30 + 40 - 20 - 15 - 10 + 5 = 80$

5. Finding Exclusive Group Membership

Q10: A survey of 200 students shows that 90 own an iPhone, 70 own a Samsung, and 50 own a Google Pixel. If 30 own both an iPhone and Samsung, 20 own both a Samsung and a Pixel, 25 own both a Pixel and an iPhone, and 10 own all three, how many own only a Samsung?

A:
$|S - (I \cup P)| = |S| - |S \cap I| - |S \cap P| + |I \cap S \cap P|$ $= 70 - 30 - 20 + 10 = 30$

6. Complement and Neither Category

Q11: If 150 people are surveyed, 90 watch Netflix, 80 watch Hulu, and 40 watch both, how many watch neither?

A:
$|N^c \cap H^c| = |U| - |N \cup H| = 150 - (90 + 80 - 40) = 150 - 130 = 20$

Q12: In a school of 300 students, 120 play football, 100 play basketball, and 80 play volleyball. If 40 play both football and basketball, 30 play both basketball and volleyball, 50 play both football and volleyball, and 20 play all three, how many play none?

A:
$|U - (F \cup B \cup V)| = 300 - (120 + 100 + 80 - 40 - 30 - 50 + 20) = 300 - 200 = 100$

7. Miscellaneous Complex Problems

Q13: If

|A| = 60

|B| = 70

|C| = 80

|A \cap B| = 25

|B \cap C| = 35

|C \cap A| = 30

, and

|A \cap B \cap C| = 15

, find

|A^c \cap B^c \cap C^c|

in a universal set of 150.

A:
$|A^c \cap B^c \cap C^c| = |U| - |A \cup B \cup C|$ $= 150 - (60 + 70 + 80 - 25 - 35 - 30 + 15) = 150 - 135 = 15$

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Thank you for taking the time to read my blog post! Your interest and engagement mean so much to me, and I hope the content provided valuable insights and sparked your curiosity. Your journey as a student is inspiring, and it’s my goal to contribute to your growth and success.

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If you found the post helpful, feel free to share it with others who might benefit. I’d also love to hear your thoughts, feedback, or questions—your input makes this space even better. Keep striving, learning, and achieving! 😊📚✨

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I recommend you check my Post on the following:

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- Jamb Mathematics- Lesson notes on Polynomials for utme Success

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This is all we can take on "Jamb Mathematics - Lesson Notes on Sets for UTME Candidate"

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SETS | Jamb Mathematics

calculation-based questions along with their answers covering different types of sets:

1. Empty Set (Null Set)

2. Universal Set

3. Complement of a Set

4. Subsets

5. Finite and Infinite Sets

6. Infinite Set

7. Disjoint Sets

8. Union and Intersection of Sets

9. Cartesian Product

10. Power Set

Calculation problems involving Cardinality of Sets

1. Basic Cardinality

2. Union of Two Sets

3. Intersection of Two Sets

4. Cardinality Using the Inclusion-Exclusion Principle

5. Complement of a Set

6. Disjoint Sets

7. Subsets and Power Set

8. Cartesian Product

9. Difference of Sets

10. Partition of a Set

Calculation problems involving not more than three (3) Sets

1. Basic Cardinality of Sets (Single Set)

2. Two-Set Problems (Union and Intersection)

3. Complement and Only-One Membership

4. Three-Set Problems (Union and Intersection)

5. Finding Exclusive Group Membership

6. Complement and Neither Category

7. Miscellaneous Complex Problems

I recommend you check my Post on the following:

POSCHOLARS TEAM.

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