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MAT 2720 Discrete Mathematics
Section 3.1 Functions
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Goals 3.1 Functions – Applications in Counting
3.3 Relations – Extension of Functions 3.4 Equivalence Relation – A special type of relation
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Goals Review and Renew the concept of functions
How to show that a function is an One-to-one function (Injection) How to show that a function is an Onto function (Surjection) Typically, you have learned these in a pre-calculus class. And it was used in calculus II for the derivatives of the inverse functions
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You Know a Lot About Functions
You are supposed to know a lot… Domain, Range, Codomain Inverse Functions One-to-one, Onto Functions Composite Functions
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Notations
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From Continuous to Discrete
Arrow Diagram Terminology:
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Is this a Function? (I)
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Is this a Function? (II)
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One-to-One Functions
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One-to-One Functions This is NOT an easy criteria to demo/prove a function is injective.
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Equivalent Criteria
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Example 1 Determine if the given function is 1-1. Prove your answer.
Proof: Analysis
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Example 2 Determine if the given function is 1-1. Prove your answer.
Proof: Analysis
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Example 2 Determine if the given function is 1-1. Prove your answer.
KEY: Must spell out the precise reasons. Since ?≠??, but f(?)=f(??), f is not injective.
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Onto Functions
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Equivalent Criteria
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Example 3 Determine if the given function is onto. Prove your answer.
Proof: Analysis
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Example 3 Determine if the given function is onto. Prove your answer.
Proof: An counter example is difficult to explain in this case Use contradiction instead – which may have the same “feel” of a counter example Analysis
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Example 4 Determine if the given function is onto. Prove your answer.
Proof: Analysis
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Template Analysis: Write down some cases to determine if a function is injective/surjective. To show injective/surjective, give a (direct) proof. To show NOT injective/surjective, use An counter example if it is easy to explain Use contradiction Other type of valid arguments
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Counting Problems…
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Counting Problems…
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Bijection
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Inverse Functions
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Group Explorations Very fun to do.
Keep the fun between you and your partner.
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