Relations and Functions Quiz

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Questions and Answers

What is true about an equivalence relation?

  • It must be transitive and injective.
  • It must be reflexive and symmetric, but not necessarily transitive.
  • It must be reflexive, symmetric, and transitive. (correct)
  • It is equivalent to an identity relation in all cases.

If set A has 4 elements, how many different relations can be formed on set A?

  • 64
  • 16
  • 256 (correct)
  • 8

Which of the following best describes a surjective function?

  • Every element in the range has a pre-image in the domain. (correct)
  • It is both injective and transitive.
  • No element in the range has a pre-image in the domain.
  • Each element in the domain has a unique corresponding element in the range.

Which of the following relations is reflexive but not symmetric?

<p>The relation that includes all pairs (a, a) and (a, b) where a ≠ b. (B)</p>
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What defines an injective function?

<p>Every element in the range has at most one pre-image in the domain. (C)</p>
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Flashcards

Relation

A subset of the Cartesian product of two sets.

Reflexive Relation

A relation where (a, a) is in the relation for every element 'a' in the set.

Identity Relation

A reflexive relation where only (a, a) pairs are present.

Symmetric Relation

If (a, b) is in the relation, then (b, a) must also be in the relation.

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Transitive Relation

If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.

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Equivalence Relation

A relation that is reflexive, symmetric, and transitive.

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Function

A relation where each input (domain) has exactly one output (range).

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Injective Function

A function where each element in the output set has at most one input.

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Surjective Function

A function where each element in the output set has at least one input.

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Bijective Function

A function that is both injective and surjective.

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Number of Relations (Finite Set)

2n2 where 'n' is the number of elements in the set.

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Number of Functions (Finite Sets)

mn , where 'n' is the number of elements in set A and 'm' is the number of elements in set B.

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Study Notes

Relations and Functions

  • A relation is a subset of the Cartesian product of two sets. The Cartesian product of sets A and B is written as A × B.

Types of Relations

  • Reflexive Relation: A relation R on a set A is reflexive if (a, a) ∈ R for every element a in A.
  • Identity Relation: A relation R on a set A is an identity relation if (a, a) ∈ R for every a in A, and no other elements are related. Basically, it only relates each element to itself.
  • Symmetric Relation: If (a, b) ∈ R, then (b, a) ∈ R. This means the relation holds both ways.
  • Transitive Relation: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. This means if a relates to b, and b relates to c, then a relates to c.

Number of Relations

  • If a set A has n elements, there are 2n2 possible relations on A.

Functions

  • A function is a specific type of relation between a domain (inputs) and a range (possible outputs).

Types of Functions

  • Injective Function (One-to-one): Every element in the range has at most one pre-image in the domain. No two different inputs map to the same output.
  • Surjective Function (Onto): Every element in the range has at least one pre-image in the domain. Every element in the codomain is hit by the function.
  • Bijective Function (One-to-one and Onto): A function that is both injective and surjective. Each output is associated with exactly one input, and every element in the range is used.

Number of Functions

  • If set A has n elements and set B has m elements, the number of functions from A to B is mn. This formula only holds for finite sets.

Equivalence Relations

  • An equivalence relation combines the properties of reflexive, symmetric, and transitive relations.

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