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 (B)

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

The relation congruence modulo m is a reflexive relation.

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

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

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

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

<p>False (B)</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

Flashcards

Congruence Relation (mod m)

A relation between positive integers where x is congruent to y modulo m (x ≡ y (mod m)) if x - y is divisible by m.

Equivalence Relation

A relation that is reflexive, symmetric, and transitive.

Reflexive Relation

A relation where every element is related to itself.

Symmetric Relation

If an element a is related to an element b, then b is related to a.

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Transitive Relation

If an element a is related to b, and b is related to c, then a is related to c.

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Congruence modulo m

A relation defined on the set of positive integers, where two integers x and y are related if their difference is a multiple of m.

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(x, y)R(u, v)

Pairs (x, y) and (u, v) are related if xy=uv in the set of ordered pairs of positive integers.

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Equivalence Class

A subset of a set that contains all elements related to a specific element under an equivalence relation.

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Symmetric Closure of R

The union of relation R and its inverse R⁻¹.

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Equivalent Elements

Elements related by an equivalence relation.

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Relation on a Set A

A set of ordered pairs from A × A.

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Length of a string

The number of characters in the string.

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Bitwise OR

A bitwise operation that produces a 1 in the output bit if at least one of the corresponding input bits is a 1.

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Bitwise AND

A bitwise operation that produces a 1 in the output bit only if both of the corresponding input bits are 1s.

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Tautology

A compound proposition that is always true, regardless of the truth values of its variables.

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Contradiction

A compound proposition that is always false, regardless of the truth values of its variables.

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Contingency

A compound proposition that can be true or false depending on the truth values of its variables.

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Logically Equivalent

Two compound propositions are logically equivalent if they have the same truth value for all possible combinations of truth values for their variables.

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Truth Table

A table that shows all possible truth values for a compound proposition, based on the truth values of its component propositions.

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Show p ≡ q

To prove that two compound propositions, p and q, are logically equivalent, you must demonstrate that their truth tables produce the same output for every possible input combination.

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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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