Podcast
Questions and Answers
What is true about an equivalence relation?
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?
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?
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?
Which of the following relations is reflexive but not symmetric?
What defines an injective function?
What defines an injective function?
Flashcards
Relation
Relation
A subset of the Cartesian product of two sets.
Reflexive Relation
Reflexive Relation
A relation where (a, a) is in the relation for every element 'a' in the set.
Identity Relation
Identity Relation
A reflexive relation where only (a, a) pairs are present.
Symmetric Relation
Symmetric Relation
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Transitive Relation
Transitive Relation
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Equivalence Relation
Equivalence Relation
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Function
Function
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Injective Function
Injective Function
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Surjective Function
Surjective Function
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Bijective Function
Bijective Function
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Number of Relations (Finite Set)
Number of Relations (Finite Set)
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Number of Functions (Finite Sets)
Number of Functions (Finite Sets)
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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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