Relations
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Questions and Answers

What is true about the relation R defined by nRm ⇔ n is a factor of m?

  • R is symmetric but not transitive
  • R is symmetric and transitive
  • R is transitive but not reflexive
  • R is reflexive but not symmetric (correct)

    • What is the minimum number of ordered pairs in a relation R on a finite set A having n elements?

  • Equal to n
  • Less than n
  • Less than or equal to n (correct)
  • Greater than n

    • What is true about a relation R on a set A if (a, b) ∈ R and (b, a) ∈ R imply R = R^(-1)?

  • R is symmetric and transitive
  • R is reflexive and transitive
  • R is an equivalence relation
  • R is reflexive and symmetric (correct)

    • What is true about a relation R on a set A if nRp and mRp imply nRp?

    <p>R is transitive</p> Signup and view all the answers

    What is true about a relation R on a set A if (a, b) ∈ R implies (b, a) ∈ R?

    <p>R is symmetric</p> Signup and view all the answers

    What is a requirement for a relation R on a set A to be reflexive?

    <p>For every a in A, (a,a) is in R</p> Signup and view all the answers

    What can be said about the identity relation on a non-void set A?

    <p>It is both reflexive and symmetric</p> Signup and view all the answers

    What is the condition for a relation R on a set A to be symmetric?

    <p>For every a, b in A, if (a,b) is in R then (b,a) is in R</p> Signup and view all the answers

    What is true about a reflexive relation on a set A?

    <p>It is not necessarily symmetric</p> Signup and view all the answers

    What is the property of the universal relation on a non-void set A?

    <p>It is both reflexive and symmetric</p> Signup and view all the answers

    What is the property of the identity relation on a set A?

    <p>It is reflexive and antisymmetric.</p> Signup and view all the answers

    If a relation R on a set A is symmetric, then what can be said about the ordered pairs (a, b) and (b, a) in R?

    <p>They are always equal.</p> Signup and view all the answers

    What is the condition for a relation R on a set A to be transitive?

    <p>For all a, b, c in A, aRb and bRc implies aRc.</p> Signup and view all the answers

    What is the diagonal line of A × A?

    <p>The set of all ordered pairs (a, b) where a = b.</p> Signup and view all the answers

    If R is a relation from A to B, which of the following is true about the domain of R?

    <p>The domain of R is a subset of A.</p> Signup and view all the answers

    Let R be a relation from a set A to itself. If (a, b) ∈ R and (b, a) ∈ R imply (a, a) ∈ R, then what can be said about R?

    <p>R is symmetric and reflexive.</p> Signup and view all the answers

    If R is the universal relation from A to B, which of the following is true about R?

    <p>R = A × B.</p> Signup and view all the answers

    If a relation R on a set A is both symmetric and reflexive, then which of the following must be true?

    <p>R is its own inverse</p> Signup and view all the answers

    Let R be a relation from a set A to itself. If for every a in A, (a, a) ∈ R, then what can be said about R?

    <p>R is reflexive but not symmetric.</p> Signup and view all the answers

    Let R be a relation from a set A to itself. If for every a, b in A, (a, b) ∈ R implies (b, a) ∈ R, then what can be said about R?

    <p>R is symmetric but not reflexive.</p> Signup and view all the answers

    What can be said about a relation R on a set A if R is reflexive and R is its own inverse?

    <p>R is symmetric but not antisymmetric</p> Signup and view all the answers

    Let R be a relation on a set A such that R is symmetric and R is antisymmetric. What can be said about R?

    <p>R is a subset of the identity relation on A</p> Signup and view all the answers

    If R is a reflexive relation on a set A, then which of the following must be true?

    <p>R is a superset of the identity relation on A</p> Signup and view all the answers

    Let R be a relation on a set A such that R is the universal relation on A. What can be said about R?

    <p>R is both symmetric and reflexive</p> Signup and view all the answers

    What can be said about the relation 'less than' in the set of natural numbers?

    <p>It is only transitive</p> Signup and view all the answers

    Which of the following relations is always symmetric?

    <p>Identity relation</p> Signup and view all the answers

    What is true about the inclusion of a subset in another, with reference to a universal set?

    <p>The relation is none of these</p> Signup and view all the answers

    What is true about the relation R defined as {(2, 4), (4, 2), (4, 6), (6, 4)} on the set A = {2, 4, 6, 8}?

    <p>The relation is only symmetric</p> Signup and view all the answers

    Which of the following statements is true about a reflexive relation R on a set A?

    <p>For every a in A, (a, a) ∈ R.</p> Signup and view all the answers

    What is the condition for a relation R on a set A to be symmetric and antisymmetric?

    <p>(a, b) ∈ R implies (b, a) ∈ R and (a, a) ∉ R.</p> Signup and view all the answers

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