Proving A and B Finite Sets: Injectivity and Surjectivity of f:A->B

  • Context: Undergrad 
  • Thread starter Thread starter algonewbee
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SUMMARY

The discussion focuses on proving properties of finite sets A and B through the function f: A -> B. It establishes that if f is injective, then the cardinality of A is less than or equal to the cardinality of B (|A| <= |B|). Conversely, if f is surjective, then the cardinality of A is greater than or equal to the cardinality of B (|A| >= |B|). The conversation emphasizes the importance of understanding the definitions of injectivity and surjectivity in the context of set theory.

PREREQUISITES
  • Understanding of set theory concepts, specifically finite sets
  • Knowledge of functions, particularly injective and surjective functions
  • Familiarity with cardinality notation, such as |A| and |B|
  • Basic mathematical proof techniques
NEXT STEPS
  • Study the definitions and properties of injective functions in set theory
  • Explore surjective functions and their implications on set cardinality
  • Learn about bijective functions and their relationship to injectivity and surjectivity
  • Review mathematical proof strategies for establishing set properties
USEFUL FOR

Students of mathematics, particularly those studying set theory, functions, and proofs; educators teaching these concepts; anyone looking to deepen their understanding of finite sets and their properties.

algonewbee
Let A and B be finite sets, and let f:A->B be a function. Show that

a)if f is injective, then |A|<=|B|
b)if f is surjective, then |A|>=|B|
 
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Are you in the wrong course? Whoever assigned these problems obviously expects that you know what "injective" and "surjective" mean! What are the definitions of those? How have you tried to apply those definitions? Do you know what it means to say that |A|= |B| for A and B sets? In particular what is the definition of |A|?
 

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