Discrete Structures - Sets

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Questions and Answers

What is the result of the operation A ∪ B if A = {5, 6} and B = {6, 7, 8}?

  • {5, 6, 7, 8} (correct)
  • {5, 6}
  • {7, 8}
  • {6, 7}

How is the symmetric difference A ⊕ B defined?

  • Elements in either A or B, but not in both (correct)
  • All elements in A excluding those in B
  • Elements in both A and B
  • Elements that are common to both A and B

What does the intersection A ∩ B yield when A = {2, 4, 6} and B = {1, 2, 3}?

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

What is the difference A − B when A = {3, 4, 5} and B = {4, 5, 6}?

<p>{3} (B)</p> Signup and view all the answers

Which of the following represents the Cartesian product of A = {1, 2} and B = {x, y}?

<p>{(1, x), (1, y), (2, x), (2, y)} (C)</p> Signup and view all the answers

If U = {1, 2, 3, 4, 5} and A = {2, 4}, what is the complement of A?

<p>{1, 3, 5} (D)</p> Signup and view all the answers

Given A = {a, b, c} and B = {b, c, d}, what is A ∩ B?

<p>{b, c} (C)</p> Signup and view all the answers

What is the result of A ⊕ B if A = {1, 2, 3} and B = {2, 3, 4}?

<p>{1, 4} (A)</p> Signup and view all the answers

What is the correct definition of a set?

<p>An unordered collection of distinct objects. (A)</p> Signup and view all the answers

Which of the following statements is true regarding the equality of sets?

<p>Two sets are equal if they have the same elements, regardless of order. (A)</p> Signup and view all the answers

What is the cardinality of the set A = {a, b, c, d}?

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

Which set represents the power set of A = {a, b}?

<p>{∅, {a}, {b}, {a, b}} (A)</p> Signup and view all the answers

Which of the following defines an infinite set?

<p>A set that is not finite. (B)</p> Signup and view all the answers

What does the notation |A| represent?

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

Which statement correctly represents the set of natural numbers?

<p>N = {1, 2, 3,...} (C)</p> Signup and view all the answers

What is a subset?

<p>A set that contains some, but not necessarily all, elements of another set. (A)</p> Signup and view all the answers

What is the result of the Cartesian product for sets A = {1, 2}, B = {a, b}, and C = {5, 6}?

<p>{&lt; 1, a, 5 &gt;, &lt; 1, a, 6 &gt;, &lt; 2, b, 5 &gt;, &lt; 2, b, 6 &gt;} (A)</p> Signup and view all the answers

Which of the following properties describes the relationship between the intersection and union of sets?

<p>A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (A)</p> Signup and view all the answers

When are two ordered n-tuples considered equal?

<p>If the sequence of elements is the same for all positions (D)</p> Signup and view all the answers

What is the result of A × ∅ if A is a non-empty set?

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

Which of the following is NOT a property of the empty set?

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

According to De Morgan's laws, which statement is true?

<p>A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (C)</p> Signup and view all the answers

What does the theorem regarding the addition principle state for finite sets A and B?

<p>|A ∪ B| = |A| + |B| - |A ∩ B| (A)</p> Signup and view all the answers

Which of the following statements about the Cartesian product is true?

<p>If either A or B is empty, then A × B = ∅ (B)</p> Signup and view all the answers

Flashcards

What is a Set?

An unordered collection of distinct objects. The objects within a set are called "elements" or "members".

What is the set of Natural Numbers (N)?

The set of natural numbers, denoted by 'N', is defined as {1, 2, 3, ...}. While some include 0, others exclude it. This set is used for counting.

What is the set of Integers (Z)?

The set of all integers, denoted by 'Z', is given by {..., −3, −2, −1, 0, 1, 2, 3, ...}. It includes all whole numbers (positive, negative, and zero).

When are two Sets equal?

Two sets are considered equal if they contain the same elements, regardless of order or repetitions.

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What is a Finite Set?

A set is considered finite if it has a limited number of distinct elements, denoted by 'n', where 'n' is a natural number. This 'n' is called the cardinality of the set and is represented by |A|.

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What is an Infinite Set?

A set is infinite if it doesn't have a limited number of elements. It contains an endless number of elements.

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What is the Power Set of a Set?

The power set of a set 'A', denoted by 'P(A)', is the set containing all possible subsets of 'A', including the empty set.

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What is the set of Rational Numbers (Q)?

The set of all rational numbers, denoted by 'Q', is defined as {p/q : p, q ∈ Z, q ≠ 0}. Rational numbers can be expressed as a fraction of two integers, where the denominator is not zero.

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Union (A ∪ B)

The set containing all elements that are in A or B.

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Intersection (A ∩ B)

The set containing all elements that are in both A and B.

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Difference (A - B)

The set containing all elements that are in A but not in B.

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Complement (AÌ…)

The set containing all elements that are not in A.

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Symmetric Difference (A ⊕ B)

The set containing elements that are in A or B, but not both.

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Ordered Pair

A pair of objects with a specific order. Written as <x, y> where x is the first element and y is the second element.

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Cartesian Product (A × B)

The set of all possible ordered pairs, formed by taking one element from set A and one element from set B.

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n-tuple

A sequence of elements that are ordered and enclosed in parentheses. For example: (a, b, c) is a 3-tuple.

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Ordered n-Tuple

A set of n objects with a specific order associated with them. It is represented as < x1, x2, ..., xn >, where x1, x2, ..., xn are the n objects.

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Cartesian Product of n Sets

The set of all possible ordered n-tuples formed by taking one element from each of n sets. It is denoted by A1 × A2 ×...× An.

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Equality of n-tuples

Two ordered n-tuples < x1, ..., xn > and < y1, ..., yn > are equal if and only if xi = yi for all i, 1 ≤ i ≤ n.

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Union of sets

The set of all elements in either A or B, or both. It is denoted by A ∪ B.

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Intersection of sets

The set of all elements that are common to both A and B. It is denoted by A ∩ B.

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Difference of sets

Theset of all elements from A that are not in B. It is denoted by A - B.

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Complement of a set

The set of all elements that are not in set A. It is denoted by A'.

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Cardinality of a set

The number of elements in a finite set. It is denoted by |A|.

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Study Notes

Discrete Structures - Sets

  • A set is an unordered collection of distinct objects.
  • Elements in a set are called members or elements.
  • Order within a set is irrelevant. {a, b, c} = {c, b, a}
  • Duplicates in a set do not change the set. {a, b, c, a} = {a, b, c}
  • The set of natural numbers (N) is {1, 2, 3,...}
  • The set of integers (Z) is {..., -3, -2, -1, 0, 1, 2, 3,...}

Sets and Subsets

  • A subset of a set A is a set containing only elements of A.
  • The power set of a set A (P(A)) is the set of all subsets of A.
    • If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}

Operations on Sets

  • Union (A ∪ B): The set containing all elements in A or B, or both.
    • A ∪ B = {x | x ∈ A or x ∈ B}
  • Intersection (A ∩ B): The set containing only elements common to both A and B.
    • A ∩ B = {x | x ∈ A and x ∈ B}
  • Difference (A – B): The set containing elements in A but not in B.
    • A – B = {x | x ∈ A and x ∉ B}
  • Complement (Ac): The set of elements in the universal set (U) but not in set A.
    • Ac = U – A
  • Symmetric Difference (A⊕B): The set of elements that are in either A or B, but not in both.
    • A⊕B = {x | (x ∈ A and x ∉ B) or (x ∈ B and x ∉ A)}

Equality of Sets

  • Two sets A and B are equal (A = B) if and only if they have the same elements.
    • ∀x[x ∈ A ⇒ x ∈ B and x ∈ B ⇒ x ∈ A]

Cartesian Product

  • The Cartesian product of two sets A and B (A × B) is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
    • Example: A = {1, 2}, B = {a, b}. A × B = {(1, a), (1, b), (2, a), (2, b)}

Properties of Sets

  • Commutative Properties: A ∪ B = B ∪ A and A ∩ B = B ∩ A
  • Associative Properties: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C
  • Distributive Properties: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  • Properties of Complement: (Ac)c = A, A ∪ Ac = U, A ∩ Ac = ∅
  • Properties of a Universal Set: A ∪ U = U, A ∩ U = A
  • Properties of the Empty Set: A ∪ ∅ = A, A ∩ ∅ = ∅

Cardinality

  • The number of distinct elements in a finite set is its cardinality (|A|)

Addition Principle

  • If A and B are finite sets, then |A ∪ B| = |A| + |B|

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