Relations

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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?

R is transitive (D) 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?

R is symmetric (C) Signup and view all the answers

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

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

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

It is both reflexive and symmetric (D) Signup and view all the answers

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

For every a, b in A, if (a,b) is in R then (b,a) is in R (B) Signup and view all the answers

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

It is not necessarily symmetric (D) Signup and view all the answers

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

It is both reflexive and symmetric (B) Signup and view all the answers

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

It is reflexive and antisymmetric. (C) 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?

They are always equal. (C) Signup and view all the answers

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

For all a, b, c in A, aRb and bRc implies aRc. (D) Signup and view all the answers

What is the diagonal line of A × A?

The set of all ordered pairs (a, b) where a = b. (B) 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?

The domain of R is a subset of A. (D) 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?

R is symmetric and reflexive. (B) Signup and view all the answers

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

R = A × B. (A) 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?

R is its own inverse (D) 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?

R is reflexive but not symmetric. (C) 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?

R is symmetric but not reflexive. (B) 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?

R is symmetric but not antisymmetric (D) 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?

R is a subset of the identity relation on A (B) Signup and view all the answers

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

R is a superset of the identity relation on A (C) 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?

R is both symmetric and reflexive (C) Signup and view all the answers

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

It is only transitive (C) Signup and view all the answers

Which of the following relations is always symmetric?

Identity relation (D) Signup and view all the answers

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

The relation is none of these (B) 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}?

The relation is only symmetric (D) Signup and view all the answers

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

For every a in A, (a, a) ∈ R. (C) Signup and view all the answers

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

(a, b) ∈ R implies (b, a) ∈ R and (a, a) ∉ R. (D) Signup and view all the answers