Set Theory, Functions. Injective/Surjective

Click For Summary
SUMMARY

The discussion focuses on the properties of functions in set theory, specifically exploring the conditions under which the composition of functions can be surjective or injective. The user seeks to construct examples of functions f and g, where f is surjective, g is injective, and their composition f*g is neither surjective nor injective. Additionally, the discussion includes a proof regarding the inverse of the composition of bijections, asserting that the inverse of the product of bijections is indeed the product of their inverses.

PREREQUISITES
  • Understanding of surjective and injective functions
  • Familiarity with function composition
  • Knowledge of bijections and their properties
  • Basic proof techniques in set theory
NEXT STEPS
  • Research examples of surjective and injective functions in ℝ and ℤ
  • Study the properties of function composition in set theory
  • Learn about the definitions and implications of bijections
  • Explore proof strategies for demonstrating properties of functions
USEFUL FOR

Students and educators in mathematics, particularly those studying set theory, functions, and their properties. This discussion is beneficial for anyone looking to deepen their understanding of injective, surjective, and bijective functions.

ktheo
Messages
51
Reaction score
0

Homework Statement



Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive

The Attempt at a Solution



I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing surjective-non-injective functions in general I tend to do them in Z since I find it easiest. I was thinking to do something involving e^x but I'm not sure. How should I approach this? Should I just think of functions I know are neither onto or one-to-one and work with products to find something?

Homework Statement



Assume that f:A→B and g:C→D are bijections. Prove that f^-1 x g^-1 is the two sided inverse of f x g (and in particular, that f x g is a bijection as well).

The Attempt at a Solution



I was wondering if someone could direct me to a similar proof or point me in the direction of some definitions that can help me here. I don't even know how to structure this into a proof.
 
Physics news on Phys.org

Similar threads

Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Show injectivity, surjectivity and kernel of groups
  • · Replies 1 ·
Replies
1
Views
2K
F bijective <=> f has an inverse
  • · Replies 7 ·
Replies
7
Views
3K
Injective & Surjective Functions
  • · Replies 9 ·
Replies
9
Views
3K
Prove that every subset ##B## of ##A=\{1,...,n\}## is finite
  • · Replies 4 ·
Replies
4
Views
1K
Set Theory: Proving h is Surjective Implies f & g
  • · Replies 3 ·
Replies
3
Views
2K
Finding a useful denial of a injective function and a surjective function
  • · Replies 3 ·
Replies
3
Views
2K
Finding the range of a function when checking if it is bijective
  • · Replies 12 ·
Replies
12
Views
2K
Prove that an endomorphism is injective iff it is surjective
  • · Replies 1 ·
Replies
1
Views
2K
Is fλ an Automorphism of the Rational Numbers Group?
  • · Replies 2 ·
Replies
2
Views
1K