Introduction to Sets in Mathematics

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Questions and Answers

What is the complement of set A = {1, 2, 3} with respect to the universal set U = {1, 2, 3, 4, 5, 6, 7}?

  • {2, 3, 4}
  • {4, 5}
  • {1, 2, 3}
  • {4, 5, 6, 7} (correct)

If set A = {1, 2, 3, 4, 5, 6} and set B = {2, 4, 6}, which statement is true?

  • B ⊆ A (correct)
  • B ∈ A
  • A ⊆ B
  • A = B

How is the symbol ∅ interpreted in relation to set A = {1, 2, 3, 4, 5, 6}?

  • ∅ is not a subset of A.
  • ∅ is an element of A.
  • ∅ is equal to A.
  • ∅ is a subset of A. (correct)

Given the sets C = {1, 2, 3} and D = {7, 8, 9}, which statement is correct?

<p>C ∩ D = ∅ (D)</p> Signup and view all the answers

Which of the following statements about the sets A = {1, 2, 3, 4, 5, 6} and D = {7, 8, 9} is valid?

<p>A ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9} (D)</p> Signup and view all the answers

What is the number of subsets for the set U = {x, y}?

<p>4 (B)</p> Signup and view all the answers

Which formula correctly calculates the number of subsets for a set with n elements?

<p>2^n (B)</p> Signup and view all the answers

If set U has 3 elements, then how many subsets does it have?

<p>8 (B)</p> Signup and view all the answers

What is the value of n(U) if U = {⌀, {x}, {y}}?

<p>3 (D)</p> Signup and view all the answers

How many subsets can be formed from the set U = {x, y, z}?

<p>8 (C)</p> Signup and view all the answers

Which of the following represents the total number of subsets for a set with 0 elements?

<p>1 (C)</p> Signup and view all the answers

For the set U = {x, y}, what are the actual subsets?

<p>{⌀, {x}, {y}, {x, y}} (D)</p> Signup and view all the answers

If n(U) = 4, how many subsets can you form?

<p>16 (D)</p> Signup and view all the answers

What is the correct notation to indicate that an object belongs to set A?

<p>x ∈ A (C)</p> Signup and view all the answers

Which of the following sets is an example of an empty set?

<p>D = {y | y is an integer greater than 10 and less than 5} (C), A = {x | x is a natural number less than 0} (D)</p> Signup and view all the answers

Which statement correctly describes a universe set?

<p>It includes all elements related to a given condition. (D)</p> Signup and view all the answers

Which of the following is a correct definition of a subset?

<p>A set that contains all elements of another set. (D)</p> Signup and view all the answers

How can a set be specified using the roster method?

<p>By listing all members separated by commas. (D)</p> Signup and view all the answers

What is the power set P(A) of a given set A?

<p>The set of all subsets of A. (D)</p> Signup and view all the answers

Which of the following represents the set of integers?

<p>ℤ (C)</p> Signup and view all the answers

What does the notation {x | P(x)} signify in set-builder notation?

<p>A set containing elements that satisfy the condition P. (C)</p> Signup and view all the answers

What is the relationship between sets A and B if A is a subset of B?

<p>B is a superset of A. (C)</p> Signup and view all the answers

Which of the following statements about subsets is always true?

<p>The null set is a subset of every set. (C)</p> Signup and view all the answers

If sets C and D are defined as C={v,x,y,z} and D={w,x,y,z}, what can be concluded about their equality?

<p>C and D are not equal. (A)</p> Signup and view all the answers

What does it imply if two sets A and B are found to be equal?

<p>A is a subset of B and B is a subset of A. (D)</p> Signup and view all the answers

How many subsets can be formed from a set U with three elements?

<p>8 (C)</p> Signup and view all the answers

Which statement about the relationships between sets is not true?

<p>A set cannot be a subset of a different set if it contains unique elements. (A)</p> Signup and view all the answers

If ℝ is defined as the set of all real numbers and ℚ is the set of rational numbers, which relationship is correct?

<p>ℝ is a superset of ℚ. (B)</p> Signup and view all the answers

What does it mean for a set to be a proper subset of another set?

<p>At least one element of the larger set is not in the smaller set. (A)</p> Signup and view all the answers

Flashcards

Set

A well-defined collection of objects.

Element of a set

An object that belongs to a set.

∈

Symbol for 'is an element of'.

∉

Symbol for 'is not an element of'.

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Empty Set (∅)

A set containing no elements.

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Universal Set (∪)

The set of all elements under consideration.

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Natural Numbers (ℕ)

The set of positive whole numbers including zero (0, 1, 2, ...).

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Integers (ℤ)

The set of whole numbers, including zero and their opposites (+/-).

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Rational Numbers (ℚ)

Numbers that can be expressed as a fraction (p/q, where p and q are integers and q ≠ 0).

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Real Numbers (ℝ)

All numbers that can be represented on a number line.

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Roster Method

Listing the elements of a set within curly braces.

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Set-builder notation

Describing the elements using a rule.

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Finite Set

A set with a specific number of elements.

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Infinite Set

A set with an unlimited number of elements.

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Subset

Every element in one set is also in another.

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Power Set

Set of all possible subsets of a given set.

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Set Equality

Two sets are equal if and only if they have exactly the same elements.

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Subset

A set A is a subset of set B (written as A ⊆ B) if every element of A is also an element of B.

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Number of Subsets

The total count of all possible subsets a set can have.

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Superset

Set B is a superset of set A (written as B ⊇ A) if every element of A is also an element of B.

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Subset of a set

A collection of one or more elements or no elements taken from a set, such as {x, y}.

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Set Equality

Two sets A and B are equal if and only if A is a subset of B and B is a subset of A (written as A = B if A ⊆ B and B ⊆ A).

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Empty Set

The set that contains no elements; it is represented by ∅.

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Cardinality of a set (n(U))

The number of elements in a set.

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Subset of Itself

Every set is a subset of itself.

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Empty Set

The empty set (∅) is a subset of every set.

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Relationship between elements in a set

Each element of the set is unordered, it is independent and has no relationship to others.

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Formula for the number of subsets

The number of subsets of a set with 'n' elements is 2^n.

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Venn Diagram

A visual representation of relationships between sets.

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Universal Set (U)

The set containing all possible elements.

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Number of Subsets

If a set has 'n' elements, it has 2n subsets.

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Complement of a Set

The set of elements in the universal set that are not in a given subset.

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Subset (⊂)

A set where all its elements are also elements of another set.

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Set Equality

Two sets are equal if they contain exactly the same elements.

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A ⊂ B

Set A is a subset of set B.

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B ⊂ A

Set B is a subset of set A.

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B ∈ C

Set B is an element of set C.

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∅ ∈ A

The empty set is an element of set A.

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∅ ⊂ A

The empty set is a subset of set A.

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Study Notes

Sets

  • A set is a well-defined collection of objects
  • Objects in a set are called elements or members
  • Sets are represented by uppercase letters (e.g., A, B, C)
  • An element belonging to a set is denoted by ∈ (e.g., a ∈ A)
  • An element not belonging to a set is denoted by ∉ (e.g., b ∉ A)
  • The empty set contains no elements (denoted by Ø)
  • The universal set (U) contains all elements relevant to a given context
  • Natural numbers (N) = {0, 1, 2, 3,...}
  • Integers (Z) = {...-2, -1, 0, 1, 2,...}
  • Rational numbers (Q)
  • Real numbers (R)
  • Power set (P(A)): the set of all subsets of set A

Specifying Sets

  • Roster method: Lists elements separated by commas (e.g., A = {1, 2, 3})
  • Set-builder notation: Describes elements using a condition (e.g., Q = {x | x is a rational number})

Finite and Infinite Sets

  • Finite set: A set with a countable number of elements (e.g., A = {1, 2, 3})
  • Infinite set: A set with an uncountable number of elements (e.g., Z)

Set Equality

  • Two sets are equal if and only if they contain exactly the same elements

Subsets

  • A subset (A ⊆ B) means every element of set A is also an element of set B
  • A set is a subset of itself (A ⊆ A)
  • The empty set (Ø) is a subset of every set (Ø ⊆ A)
  • If A ⊆ B and B ⊆ A, then A = B (sets are equal)

Set Relations

  • If every element of Set A is also an element of Set B, then A is a subset of B (A ⊆ B)
  • The relationship of a set being a subset of another is often denoted by the symbol ⊆.
  • The symbol ⊂ denotes proper subset, meaning A is a subset of B but not equal to B.

Set Operations

  • Complement (A'): The set of elements in the universal set (U) that are not in set A. (A' = {x ∈ U | x ∉ A})
  • Union (A ∪ B): The set of all elements that are in A or B or both.
  • Intersection (A ∩ B): The set of all elements that are in both A and B.
  • Difference (A - B): The set of elements in A but not in B.
  • Symmetric difference (A Δ B): The set of elements that are in A or B, but not in both.

Subset Formation

  • The total number of subsets of a set with 'n' elements is 2n.

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Discrete Structures 1 - Module 6: Sets - PDF

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