Logic and Equivalence Relations Quiz
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Questions and Answers

Which of the following is an example of a tautology?

  • p ∨ ¬p (correct)
  • p ∧ q
  • p ∧ ¬p
  • p ∨ q

    • A contradiction is a compound proposition that can sometimes be true.

    False

    What is the term for a compound proposition that is neither a tautology nor a contradiction?

    contingency

    The notation used to denote that two compound propositions are logically equivalent is _____.

    <p>p ≡ q</p> Signup and view all the answers

    What does the notation x ≡ y (mod m) signify?

    <p>x − y is divisible by m</p> Signup and view all the answers

    The relation congruence modulo m is a reflexive relation.

    <p>True</p> Signup and view all the answers

    Match the terms with their definitions:

    <p>Tautology = Always true Contradiction = Always false Contingency = Neither always true nor always false Logically Equivalent = p ↔ q is a tautology</p> Signup and view all the answers

    Which of the following pairs of propositions are logically equivalent?

    <p>p → q and ¬p ∨ q</p> Signup and view all the answers

    List the three properties that make a relation an equivalence relation.

    <p>Reflexivity, Symmetry, Transitivity</p> Signup and view all the answers

    The relation R defined by (x,y)R(u,v) if and only if xv = yu is ___.

    <p>an equivalence relation</p> Signup and view all the answers

    The expression (p ∨ q) ↔ (q ∨ p) is an example of a tautology.

    <p>True</p> Signup and view all the answers

    Match the property of equivalence relations to its definition:

    <p>Reflexivity = For all x, x is related to x Symmetry = If xRy, then yRx Transitivity = If xRy and yRz, then xRz</p> Signup and view all the answers

    What is the result of the bitwise AND operation between the bit strings 1010 and 1100?

    <p>1000</p> Signup and view all the answers

    Which of the following is an example of a congruence relation?

    <p>x and y have the same remainder when divided by m</p> Signup and view all the answers

    In an equivalence class, every element is related to every other element in that class.

    <p>True</p> Signup and view all the answers

    Provide an example of two integers that are congruent modulo 5.

    <p>83 and 13</p> Signup and view all the answers

    Which of the following properties must a relation possess to be classified as an equivalence relation?

    <p>Reflexive, Symmetric, Transitive</p> Signup and view all the answers

    The relation R defined by aRb if and only if a − b is an integer is an equivalence relation.

    <p>True</p> Signup and view all the answers

    What notation is often used to denote that two elements are equivalent with respect to a particular equivalence relation?

    <p>a ∼ b</p> Signup and view all the answers

    The symmetric closure of a relation R is defined as R ∪ R⁻¹, which consists of pairs (a, b) such that _____.

    <p>a is related to b or b is related to a</p> Signup and view all the answers

    In the equivalence relation where aRb if and only if l(a) = l(b), what does l(x) represent?

    <p>Length of the string x</p> Signup and view all the answers

    If aRb and bRc for an equivalence relation, it does not necessarily follow that aRc.

    <p>False</p> Signup and view all the answers

    List the properties an equivalence relation must possess.

    <p>Reflexivity, Symmetry, Transitivity</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Reflexivity = Relation where every element is related to itself Symmetry = If aRb, then bRa Transitivity = If aRb and bRc, then aRc Equivalence Class = Set of elements that are all equivalent to each other under a specific relation</p> Signup and view all the answers

    Study Notes

    Sets

    • A set is an unordered collection of objects.
    • The objects in a set are called elements or members.
    • A ∈ B means that a is an element of set B
    • A ∉ B means a is not an element of set B

    Set Operations

    • Union (∪): The union of two sets A and B, denoted by A ∪ B, is the set containing all elements that are in A, in B, or in both.
    • Intersection (∩): The intersection of two sets A and B, denoted by A ∩ B, is the set containing only the elements that are common to both A and B.
    • Difference (–): The difference of set A and B, denoted by A – B, is the set containing elements that are in A but not in B.
    • Complement (¯): The complement of A, denoted by Ā, is the set containing elements in the universal set U but not in A.

    Types of Sets

    • Universal Set (U): The universal set is the set of all objects under consideration.
    • Empty Set (Ø or {} ): The empty set is the set that contains no elements.
    • Singleton Set: A set containing only one element.

    Power Set

    • The power set of a set S (denoted as P(S)) is the set of all possible subsets of S.

    Cartesian Product

    • The Cartesian product of sets A and B (denoted as A × B) is the set of all possible ordered pairs where the first element of each pair is from A and the second element from B.

    Ordered Pairs

    • These are elements that have a particular sequence or order, different from sets where their order does not matter

    Relations

    • A binary relation from A to B is a subset of A x B, where the ordered pairs (a,b) where a is in A and b is in B.
    • Reflexive: For all a ∈ A, then (a, a) ∈ R.
    • Symmetric: If (a, b) ∈ R, then (b, a) ∈ R.
    • Antisymmetric: If (a, b) ∈ R and (b, a) ∈ R, then a = b.
    • Transitive: If(a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

    Equivalence Relations

    • An equivalence relation is a relation that is reflexive, symmetric, and transitive.

    Equivalence Classes

    • An equivalence class of an element a in a set A is the set of all elements in A related to a.

    Functions

    • A function f from A to B is a relation where each element in A is associated with one and only one element in B.

    • Domain: The set of all inputs (elements of A) to a function.

    • Codomain: The set of all possible outputs (elements of B) of a function.

    • Range: The actual set of outputs (elements of B) of the function, when given the domain.

    • One to one (injective): Every element in the codomain is mapped to by at most one element in the domain.

    • Onto (surjective): Every element in the codomain is mapped to by at least one element in the domain.

    • Bijective: A function that is both one-to-one and onto.

    Inverse Function

    • The inverse function of f (denoted by f⁻¹) reverses the relation of f(x) to create f⁻¹(y) = x. It reverses the input and output relationships.

    Composition of Functions

    • The composition of functions is defined as (f∘g)(x) = f(g(x)).

    Floor and Ceiling Functions

    • The floor function ([x]) gives the greatest integer less than or equal to x.
    • The ceiling function ([x]) gives the least integer greater than, or equal to x.

    Pigeonhole Principle

    • If you have more objects than boxes, at least one box will have to contain two or more items.

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    Related Documents

    MAT2004 Complete Notes PDF

    Description

    Test your understanding of tautologies, equivalence relations, and logical equivalence. This quiz covers concepts such as congruence relations and bitwise operations. Dive into the realm of symbolic logic and evaluate your knowledge!

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