Sets Quiz 2020-21

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Questions and Answers

Which description correctly represents the set A of natural number squares?

A = {x : x is an integer and x > 0}

A = {x : x is a natural number}

A = {x : x = n, where n ∈ N}

A = {x : x = n^2, where n ∈ N} (correct)

In set-builder notation, how would you write the set of fractions from 1/2 to 6/7?

x : x = 1/n, where n is a natural number and n ≤ 6

x : x = n/(n-1), where n is any integer less than 6

x : x = (n-1)/n, where n is a natural number and 1 ≤ n ≤ 6 (correct)

x : x = n/n+1, where n is a natural number and 1 ≤ n ≤ 6

Which set matches the description 'x is a letter of the word PRINCIPAL'?

A = {P, R, I, N, C, A, L, I}

A = {P, R, I, N, C, A, L} (correct)

A = {x : x is a letter of the alphabet repeated}

A = {x : x is in the alphabet}

What set matches the condition 'x is a positive integer and is a divisor of 18'?

{1, 2, 3, 6, 9, 18} Signup and view all the answers

Which equation correctly represents x if x + 1 = 1?

x = 0 Signup and view all the answers

Which of the following collections can be considered a set?

The collection of all natural numbers less than 100. Signup and view all the answers

What is the result of inserting the appropriate symbol in the blank space for 4...A if A = {1, 2, 3, 4, 5, 6}?

4 ∈ A Signup and view all the answers

Which expression reflects the collection of all even integers?

{x : x = 2n, where n ∈ Z} Signup and view all the answers

Is the set of lines which are parallel to the x-axis finite or infinite?

Infinite Signup and view all the answers

How many letters are in the English alphabet?

26 Signup and view all the answers

Which of the following sets is finite?

The set of numbers which are multiples of 5 Signup and view all the answers

Are the sets A = {a, b, c, d} and B = {d, c, b, a} equal?

Yes, they are equal Signup and view all the answers

Given A = {4, 8, 12, 16} and B = {8, 4, 16, 18}, are these sets equal?

No, they are not equal Signup and view all the answers

Which of the following expressions correctly denotes that set Y is a subset of set X?

Y ⊂ X Signup and view all the answers

Identify which pairs of sets are equal: A = {2, 4, 8, 12} and C = {4, 8, 12, 14}.

They are not equal due to differing elements Signup and view all the answers

Which statement is true regarding subsets?

A set can be a subset of itself Signup and view all the answers

What did Bertrand Russell demonstrate regarding the set of all sets?

Its existence leads to a contradiction. Signup and view all the answers

Which mathematician published the first axiomatisation of set theory?

Ernst Zermelo Signup and view all the answers

What principle did Gottlob Frege attribute to set theory at the turn of the century?

It is rooted in the principles of logic. Signup and view all the answers

Which mathematician introduced the axiom of regularity?

John Von Neumann Signup and view all the answers

What does Paul R. Halmos state about containment in set theory?

Nothing contains everything. Signup and view all the answers

Which of the following represents set A in roster form?

{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6} Signup and view all the answers

Which of the following natural numbers is included in set B?

5 Signup and view all the answers

What is the sum of the digits of the two-digit natural numbers in set C?

8 Signup and view all the answers

Which of the following numbers is included in set D?

2 Signup and view all the answers

Which set represents all letters in the word TRIGONOMETRY?

{T, R, I, G, O, N, M, E, Y} Signup and view all the answers

Which of the following correctly represents set F in roster form?

{B, E, T, R} Signup and view all the answers

Which of the following sets is an example of a null set?

{x : x is a natural number, x < 5 and x > 7} Signup and view all the answers

Which of the following sets is infinite?

{1, 2, 3,...} Signup and view all the answers

What does the union of two sets A and B represent?

The set of all elements that are in either A or B. Signup and view all the answers

Which law describes the property A ∪ B = B ∪ A?

Commutative Law Signup and view all the answers

If sets A and B are disjoint, what is the value of A ∩ B?

φ (empty set) Signup and view all the answers

What does the symbol ‘∩’ denote in set theory?

The intersection of two sets. Signup and view all the answers

Which property states that (A ∪ B) ∪ C = A ∪ (B ∪ C)?

Associative Law Signup and view all the answers

If A = {2, 4, 6, 8} and B = {1, 3, 5, 7}, how would one describe A and B?

They are disjoint sets. Signup and view all the answers

Which law states that A ∪ φ = A?

Identity Law Signup and view all the answers

For sets A and B, if A ∩ B = B, what can be inferred about the relationship between A and B?

A is a subset of B. Signup and view all the answers

Study Notes

Set Definitions and Examples

Set A consists of squares of natural numbers: A = {x : x is the square of a natural number}, alternatively written as A = {x : x = n², where n ∈ N}.
For the set {2, 3, 4, 5, 6, 7}, its set-builder form is x = n/(n + 1), where n is a natural number and 1 ≤ n ≤ 6.

Matching Sets

Set {P, R, I, N, C, A, L} matches the set of letters in "PRINCIPAL" since it includes the same letters.
{0} matches the equation x + 1 = 1, yielding x = 0.
The divisors of 18 correspond to {1, 2, 3, 6, 9, 18}.
The equation x² - 9 = 0 yields the solutions x = 3, -3.

Identifying Sets

Examples of sets include:
- Months of a year starting with 'J' is a valid set.
- "Ten most talented writers of India" is subjective, thus not a well-defined set.
- "A team of eleven best-cricket batsmen of the world" is subjective and varies.
- "All boys in your class" is a specific subset.
- "Natural numbers less than 100" is a valid set.
- "Novels by Munshi Prem Chand" is also a well-defined set.
- "All even integers" is a valid infinite set.
- Collections like chapter questions or dangerous animals are subjective.

Set Membership

Indicate membership with ∈ or ∉:
- 5 ∈ A, 8 ∉ A, 0 ∉ A, 4 ∈ A, 2 ∈ A, 10 ∉ A.

Roster Form Creation

Roster forms of given sets include:
- A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6} for integers between -3 and 6.
- B = {1, 2, 3, 4, 5} for natural numbers less than 6.
- C = {17, 26, 35, 44, 53} for two-digit natural numbers summing to 8.
- D = {2, 3, 5, 15} for prime divisors of 60.
- E = {T, R, I, G, O, N, M, E, T, R, Y} from "TRIGONOMETRY."
- F = {B, E, T, R} from "BETTER."

Set-Builder Form Creation

Set-builder forms for specified sets include:
- {3n | n ∈ N} for {3, 6, 9, 12}.
- {2^n | n ∈ N} for {2, 4, 8, 16, 32}.
- {5^n | n ∈ N} for {5, 25, 125, 625}.
- {2n | n ∈ N} for {2, 4, 6, ...}.
- {n² | n ∈ N and n² ≤ 100} for {1, 4, 9, ..., 100}.

Elements of Sets

Elements of specified sets include:
- A = {1, 3, 5, 7, ...} for odd natural numbers.
- B = {-4, -3, -2, -1, 0, 1, 2, 3, 4} for integers below 5.
- C = {5} for solutions to x - 5 = 0.
- D = {-5, 5} for x² = 25.
- E = {5} for positive integral roots of the equation x² - 2x - 15 = 0.

Set Equality and Comparison

Sets A and B are equal if they contain the same elements, including order and repetition.
Example: Set of letters in "ALLOY" equals the set of letters in "LOYAL" since both contain A, L, O, Y.

Null Sets

Examples of null sets include:
- Set of odd natural numbers divisible by 2 is empty.
- Even prime numbers yield the null set as only 2 is even.
- {x | x is a natural number, x < 5 and x > 7} is empty.

Finite and Infinite Sets

Finite sets include:
- Months of the year, which contains 12 members.
- Natural numbers ranging from 1 to 100 is finite.
Infinite sets include:
- {1, 2, 3,...} and positive integers greater than 100.
- Prime numbers below 99 vary infinitely.

Subsets

A subset is defined as a set A where all elements are contained in set B (A ⊆ B).
Subset notation employs "⊆," meaning all members of A are also in B, while "⊄" denotes A is not a subset of B.

Union and Intersection of Sets

Union (A ∪ B) combines elements from both sets, including common elements.
Properties of union:
- A ∪ B = B ∪ A (commutative).
- Associative property allows grouping without affecting results.
- The union of a set and the empty set retains the original set.
Intersection (A ∩ B) includes common elements from both sets, denoted symbolically.

Disjoint Sets

Disjoint sets have no common elements, such as A = {2, 4, 6, 8} and B = {1, 3, 5, 7}.

Historical Context in Set Theory

Russell's Paradox highlighted contradictions in set theory assumptions.
Ernst Zermelo laid foundational axioms to resolve these paradoxes.
Ongoing modifications and developments in set theory continue to advance mathematical language and understanding.

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Description

Test your understanding of finite and infinite sets with this quiz. Evaluate various sets from letters to animals and numbers. Determine whether each set listed is finite or infinite based on their nature.