Sets Quiz 2020-21

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.

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.