Function Equivalence: Proving Equality of Functions in F(S,F)

In summary, two functions in F(S,F) are equal if and only if they have the same value at each element of S. To prove that if two functions are equal, they have the same value at each element of S, we need to use the definition of equivalence and the definition of a function to show that all parts of the functions are the same.
  • #1
Seacow1988
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Homework Statement



Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False?

Homework Equations



How can you prove: if two functions, x and y, are equal then they have the same value at each element of S?

The Attempt at a Solution



By the definition of equivalence, I can see that if two functions have the same value at each element of S, they are equal. However, I'm not sure how to show the converse.
 
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  • #2
Seacow1988 said:

Homework Statement



Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False?

Homework Equations



How can you prove: if two functions, x and y, are equal then they have the same value at each element of S?

The Attempt at a Solution



By the definition of equivalence, I can see that if two functions have the same value at each element of S, they are equal. However, I'm not sure how to show the converse.
What is the definition of "equal" for functions? Typically, two mathematical "objects" are said to be "equal" if all parts of them are the same. Okay, what is the definition of "function"? What "parts" does a function have?
 

FAQ: Function Equivalence: Proving Equality of Functions in F(S,F)

What is function equivalence?

Function equivalence is a mathematical concept that refers to two functions being equal in terms of their input-output relationships. This means that for every input value, both functions produce the same output value.

How is function equivalence proven?

Function equivalence is typically proven using a series of logical steps and mathematical properties. This can involve simplifying both functions and showing that they are equivalent through substitution and manipulation.

Why is function equivalence important?

Function equivalence is important because it allows us to determine if two seemingly different functions are actually the same in terms of their behavior. This can help in simplifying complex functions and understanding the relationship between different mathematical expressions.

What is the role of sets and functions in function equivalence?

In order to prove function equivalence, we must first define the sets and functions involved. Sets provide the domain and range for the functions, while the functions themselves dictate the relationship between the elements of the sets.

Are there any common mistakes when proving function equivalence?

Yes, some common mistakes when proving function equivalence include not considering the domains and ranges of the functions, assuming that two functions are equivalent without proper proof, and making incorrect substitutions or manipulations. It is important to carefully follow logical steps and ensure that all properties and rules are applied correctly.

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