SET THEORY: BASIC TO ADVANCED QUIZ 1

SUPPOSE WE'RE WORKING WITH NATURAL NUMBERS LESS THAN 10. THEN:

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

LET:

A={2,4,6,8} (EVEN NUMBERS)

B={1,3,5,7,9} (ODD NUMBERS)

HERE, BOTH A AND B ARE SUBSETS OF THE UNIVERSAL SET U.

total subsets = 2⁴=2*2*2*2=16

We can solve this using the principle of inclusion and exclusion.

Let:

• Y be the number of people who use YouTube = 70

• I be the number of people who use Instagram = 50

• Y∩IY be the number who use both = 30

The number of people who use at least one (YouTube or Instagram) is:

Y∪I=Y+I−(Y∩I)

Y∪I=70+50−30=90

Answer: 90 people use at least one of them.

We are given:

• People who like tea = 60

• People who like coffee = 40

• People who like both = 20

We are to find how many people like only tea or only coffee.

Step 1: Use the principle of inclusion and exclusion

Let:

• T = set of people who like tea

• C = set of people who like coffee

• T∩C = people who like both = 20

• Then:

  • Only tea = T−(T∩C)=60−20=40

  • Only coffee = C−(T∩C)=40−20=20

  • Step 2: Add only tea and only coffee

    $$=60$$

  • Final Answer:

    60 people like only tea or only coffee

GIVEN:

• LIKE TENNIS = 70
• LIKE FOOTBALL = 50
• LIKE BOTH = 30

PEOPLE WHO LIKE ONLY TENNIS:

ONLY TENNIS=LIKE TENNIS−LIKE BOTH=70−30=40=70−30=40

ANSWER: 40 PEOPLE LIKE ONLY TENNIS.

The symbol ⊂ typically means proper subset, i.e.,
A⊂B implies every element of A is in B, but A≠B.

• Therefore, saying A=B contradicts the definition of a proper subset.

Example:

Let A={1,2} and B={1,2,3}.
Then A⊂B (proper subset),
but clearly A≠B, so the statement is false.

Most likely interpretation (for elementary set theory level):

Assume:

• A={1,2}

• B={{1,2} is not valid But maybe it was a typo or meant to be:

• B={{1,2},"B"} — two elements: the set {1,2} and the symbol "B"

• Then:

  • ∣A∣=2

  • ∣B∣=2

  • ∣A×B∣=∣A∣⋅∣B∣=2×2=4

  • Final Answer:

    If $A=\{1,2\}$ and B={{1,2},B}, then B is not a well-defined set under standard set theory.

    But if B has 2 elements, then:

    ∣A×B∣=4

∣P(A)∣=2ⁿ

IF THE SET HAS 3 ELEMENTS, THEN:

∣P(A)∣=2³=2*2*2=8

Given:

• Total students = 100

• Students who like Movies (M) = 80

• Students who like Music (Mu) = 60

• Students who like both = 50

• Step 1: Find students who like either movies or music
• ∣M∪Mu∣=∣M∣+∣Mu∣−∣M∩Mu∣=80+60−50=90
• Step 2: Find students who like neither
• Neither=Total−∣M∪Mu∣=100−90=10​
• Final Answer: 10 students like neither movies nor music.

Step 1: Identify vowels in “education”

The word education is made up of the letters:
e, d, u, c, a, t, i, o, n

Now, the vowels in the English alphabet are:
a, e, i, o, u

So, the vowels in "education" are:
e, u, a, i, o

Step 2: Write them in set notation (no repetition, order doesn’t matter in sets)

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

Final Answer:

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

Solution:
only math = 45 – 20 = 25
only science = 30 – 20 = 10
total = 35

We are given:

• Football (F) = 70

• Basketball (B) = 60

• Volleyball (V) = 50

• Football ∩ Basketball = 30

• Basketball ∩ Volleyball = 20

• Football ∩ Volleyball = 25

• All three (F ∩ B ∩ V) = 10

• We are to find the number of students who play at least one sport, i.e., the union:

  ∣F∪B∪V∣

• Step 1: Use the formula for three-set union

  ∣F∪B∪V∣=∣F∣+∣B∣+∣V∣−∣F∩B∣−∣B∩V∣−∣F∩V∣+∣F∩B∩V∣

• Plug in the values:

  ∣F∪B∪V∣=70+60+50−30−20−25+10

• =180−75+10=115

• Final Answer:

  115 students play at least one sport

Given:

• Total students = 40

• Students who like English (E) = 28

• Students who like Math (M) = 22

• Students who like both = 14

• Step 1: Find students who like only English
• Only English=∣E∣−∣E∩M∣=28−14=14

• Step 2: Find students who like only Math

  Only Math=∣M∣−∣E∩M∣=22−14=8

• Step 3: Add both
• Only one subject=14+8=22​
• Final Answer: 22 students like only one of the subjects.

A NULL SET (OR EMPTY SET) IS A SET WITH NO ELEMENTS, DENOTED BY ∅ OR {}.
A SINGLETON SET IS A SET WITH EXACTLY ONE ELEMENT, E.G., {5}.

We are given:

• Total students = 100

• Students who like Mathematics = 80

• Students who like Physics = 70

• Students who like both = 60

We are to find how many students like only Mathematics or only Physics (not both).

Step 1: Define sets

Let:

• M = students who like Mathematics = 80

• P = students who like Physics = 70

• M∩P = students who like both = 60

• Then:

  • Only Mathematics = M−(M∩P)=80−60=20

  • Only Physics = P−(P∩M)=70−60=10

  • Step 2: Add them up

    Only Mathematics or Only Physics=20+10=30

  • Final Answer:

    30 students like only Mathematics or only Physics

FOR ANY SET A, ALL ELEMENTS OF A ARE IN A, SO A⊆A

We are given:

• % of students who take Math = 40%

• % who take Biology = 30%

• % who take both = 10%

We are to find the percentage who take either Math or Biology, i.e., the union of the two sets.

Step 1: Use the formula for union of two sets

P(Math ∪ Biology)=P(Math)+P(Biology)−P(Both)

=40%+30%−10%=60%

Final Answer:

60% of students take either Math or Biology

THERE ARE NO COMMON ELEMENTS BETWEEN A AND B.

o find the number of subsets of a set, use the formula:

Number of subsets= 2ⁿ

where n is the number of elements in the set.

Given set: {0,1,2}

where n is the number of elements in the set.

It has n=3 elements.

Number of subsets=2³=8​

Final Answer: 8 subsets

THE NULL SET (ALSO CALLED THE EMPTY SET, DENOTED ∅) IS A SUBSET OF EVERY SET BECAUSE THERE ARE NO ELEMENTS IN IT TO VIOLATE SUBSET CONDITIONS.

SOLUTION:
ONLY CHESS = 20 − 10 = 10
ONLY CARROM = 30 − 10 = 20

TOTAL = 10 + 20 = 30

We’ll use the principle of inclusion and exclusion again, and subtract from the total to find how many play neither.

Let:

• Total students = 50

• Football players = 30

• Cricket players = 25

• Both = 10

• First, find how many play at least one (Football or Cricket):
• At least one=30+25−10=45

• Now subtract from the total to find how many play neither:

  Neither=50−45=5

• Answer: 5 students play neither.

WE ARE GIVEN:

TOTAL STUDENTS = 30

STUDENTS WHO STUDY MATH = 18

STUDENTS WHO STUDY SCIENCE = 12

STUDENTS WHO STUDY BOTH MATH AND SCIENCE = 5

WE WANT TO FIND HOW MANY STUDENTS STUDY EITHER MATH OR SCIENCE. THAT MEANS:

∣M∪S∣= |M| + |S| - ∣M∩S∣

SUBSTITUTE THE VALUES:

∣M∪S∣=18+12−5=25

SOLUTION:
ONLY TEA = 65 − 20 = 45

We can use the principle of inclusion and exclusion again.

Given:

• Total students = 40

• Like Math = 28

• Like Science = 20

• Like both = 8

• irst, find how many like at least one (Math or Science):

  At least one=28+20−8=40

  Now subtract from total to find how many like neither:

  Neither=40−40=0

• Answer: 0 students like neither. Everyone likes at least one subject.

We are given:

• People who like Cricket = 45

• People who like Football = 25

• People who like both = 10

We are to find how many people like only Cricket or only Football.

Step 1: Calculate people who like only one sport

• Only Cricket = 45 − 10 = 35

• Only Football = 25 − 10 = 15

• Step 2: Add both

  Only Cricket or Only Football=35+15=50

• Final Answer:

  50 people like only Cricket or only Football

Step 1: Find A∩B

This is the set of elements common to both A and B:

A∩B={2,3}

Step 2: Find B∩C

This is the set of elements common to both B and C:

B∩C={3,4,5}

Step 3: Take the union of the two results

(A∩B)∪(B∩C)={2,3}∪{3,4,5}

Combine all unique elements:

{2,3,4,5}

Final Answer:

(A∩B)∪(B∩C)={2,3,4,5}

Given:

• Total people = 100

• Like apples = 60

• Like oranges = 45

• Like both apples and oranges = 25

• Step 1: People who like either apples or oranges (or both):

  ∣A∪O∣=∣A∣+∣O∣−∣A∩O∣=60+45−25=80

• Step 2: People who like neither:

  Neither=Total−∣A∪O∣=100−80=20

• 20 people like neither apples nor oranges.

ANSWER:

• TOTAL SUBSETS = 2⁴=16
• NON-EMPTY = 16−1=15

We are given:

• Total people = 80

• Like Tea = 50

• Like Coffee = 40

• Like Both = 20

We are to find how many people like only Tea or only Coffee (i.e., not both).

Step 1: Find people who like only Tea:

Only Tea=50−20=30

Step 2: Find people who like only Coffee:

Only Coffee=40−20=20

Step 3: Add them together:

Only Tea or Only Coffee=30+20=50

Final Answer:

50 people like only Tea or only Coffee

What is the empty set?

∅={}

It contains no elements.

Power set definition:

The power set of a set is the set of all its subsets, including the empty set and the set itself.

Subsets of the empty set:

The only subset of the empty set is... the empty set itself!

P(∅)={∅}

So, how many elements in the power set?

P(∅)∣=1

Final Answer:

Yes, the power set of an empty set contains exactly one element:

{∅}​

We are given:

• Total students = 60

• Like Maths = 25

• Like Science = 30

• Like both = 10

We are to find how many students like neither Maths nor Science.

Step 1: Use the formula for union of two sets

∣M∪S∣=∣M∣+∣S∣−∣M∩S∣=25+30−10=45

So, 45 students like either Maths, Science or both.

Step 2: Subtract from total to find those who like neither

Neither=60−45=15

Final Answer:

15 students like neither Maths nor Science

We are given:

• Total people = 100

• People who speak English = 45

• People who speak Hindi = 60

• People who speak both languages = 25

We are to find how many people speak neither English nor Hindi.

Step 1: Use the formula for union of two sets:

∣E∪H∣=∣E∣+∣H∣−∣E∩H∣

∣E∪H∣=45+60−25=80

So, 80 people speak either English, Hindi, or both.

Step 2: Subtract from total to find who speak neither:

Neither=100−80=20

Final Answer:

20 people speak neither English nor Hindi​

Solution:

• Only cricket = 60 – 25 = 35
• Only hockey = 40 – 25 = 15
  total = 50

A singleton set is a set with exactly one element.
Let’s say the singleton set is:

A={x}

Step 1: Recall the formula for the cardinality of a power set:

If a set has $n$ elements, then its power set has 2ⁿ
elements.

Here, $1$ (since it's a singleton set).

So,

∣P(A)∣=

2¹

$$=2$$

Step 2: What are the elements of the power set?

P(A)={∅,{x}}

Final Answer:

2​

We are given:

• Total students = 60

• Like Maths = 25

• Like Science = 30

• Like both = 10

We are to find how many students like neither Maths nor Science.

Step 1: Use the formula for union of two sets

∣M∪S∣=∣M∣+∣S∣−∣M∩S∣=25+30−10=45

So, 45 students like either Maths, Science or both.

Step 2: Subtract from total to find those who like neither

Neither=60−45=15

Final Answer:

15 students like neither Maths nor Science

THE NUMBER OF PROPER SUBSETS OF A SET WITH NNN ELEMENTS IS:

2^N−1

THIS IS BECAUSE THE TOTAL NUMBER OF SUBSETS IS 2^N, AND A PROPER SUBSET EXCLUDES THE SET ITSELF.

FOR A SET WITH 5 ELEMENTS:

2⁵−1=32−1=31

ANSWER: 31 PROPER SUBSETS

We are given:

• Total students = 100

• Like Chocolate = 75

• Like Vanilla = 50

• Like Both = 30

We are to find how many students like only one flavor (i.e., only Chocolate or only Vanilla, not both).

Step 1: Find students who like only Chocolate

Only Chocolate=75−30=45

Step 2: Find students who like only Vanilla

Only Vanilla=50−30=20

Step 3: Add them

Only one flavor=45+20=65

Final Answer:

65 students like only one flavor​

Given:

• Total students = 50

• Students who play cricket (C) = 30

• Students who play football (F) = 25

• Students who play both cricket and football (C ∩ F) = 10

• Step 1: Find students who play either cricket or football

  Using inclusion-exclusion:

• ∣C∪F∣=∣C∣+∣F∣−∣C∩F∣=30+25−10=45

• Step 2: Find students who play neither

  Neither=Total students−∣C∪F∣=50−45=5

• Final Answer: 5 students play neither cricket nor football.

We are given:

• Total students = 60

• Like Maths = 25

• Like Science = 30

• Like both = 10

We are to find how many students like neither Maths nor Science.

Step 1: Use the formula for union of two sets

∣M∪S∣=∣M∣+∣S∣−∣M∩S∣=25+30−10=45

So, 45 students like either Maths, Science or both.

Step 2: Subtract from total to find those who like neither

Neither=60−45=15

Final Answer:

15 students like neither Maths nor Science

2×2×2=8

• a ∪ b = {1,2,3,4}
• a ∩ (a ∪ b) = a
  answer: {1,2,3}