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

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SUMMARY

In the context of functions in F(S,F), two functions are equal if and only if they yield the same output for every input in the vector space S. This conclusion is supported by the definition of function equality, which states that all components of the functions must match. The discussion emphasizes the necessity of proving the converse: that if two functions are equal, they must have identical values at each element of S. Understanding these principles is crucial for mathematical rigor in function analysis.

PREREQUISITES
  • Understanding of vector spaces, specifically F(S,F)
  • Knowledge of function definitions and properties
  • Familiarity with mathematical proofs and equivalence relations
  • Basic concepts of mathematical logic
NEXT STEPS
  • Study the properties of equivalence relations in mathematics
  • Learn about the formal definitions of functions and their components
  • Explore proofs involving function equality and implications
  • Investigate examples of functions in vector spaces to solidify understanding
USEFUL FOR

Students of mathematics, particularly those studying linear algebra or functional analysis, as well as educators seeking to clarify concepts of function equivalence and equality.

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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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?
 

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