How to prove injection and surjection for a function with 2 variables?

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SUMMARY

The discussion focuses on proving injection and surjection for functions of two variables, specifically the function f: R x R → R defined by f(x, y) = x + y. It is established that this function is not injective, as demonstrated by the example f(1, 0) = f(0, 1). However, the function is surjective since for any real number a, there exists a pair (a, 0) such that f(a, 0) = a.

PREREQUISITES
  • Understanding of functions and their properties, specifically injection and surjection.
  • Familiarity with the Cartesian product of sets, particularly R x R.
  • Basic knowledge of real numbers and their operations.
  • Experience with function notation and mapping concepts.
NEXT STEPS
  • Study the definitions and properties of injective and surjective functions in depth.
  • Explore examples of functions with two variables and their mappings.
  • Learn about the implications of injectivity and surjectivity in higher-dimensional functions.
  • Investigate the role of function composition in determining injective and surjective properties.
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Mathematics students, educators, and anyone interested in understanding the properties of functions, particularly in the context of multivariable calculus and analysis.

pmooney12
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how do you prove injection and surjection of the function of 2 variables. for example f:RxR->R
 
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The same way you prove it for 1 variable.
Can you give us a specific map?
 
For example the map f:RxR--> R:x-->x+y.

This is not an injection, since f(1,0)=f(0,1).
This is a surjection. Take a in R, then f(a,0)=a.
 

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