• Chapter 2
  • Function
    • Let A and B are set. If every element in A has assignment of exactly one element in B, this assignment is a 'function'
    • Denotion: f: A -> B
    • A: Domain, B: Codomain
    • Image
      • Assigned target of an element in domain. Image is in codomain
      • Denotion: For example, image of a is f(a)
    • Range: Set of images of all element in domain
    • Types
      • onto function(surjective)
        • Definition: Every element in codomain has preimage
        • (range of image) = (codomain)
      • one-to-one function(injective)
        • Definition: (f(a) = f(b)) -> a = b
      • One-to-one correspondence(Bijection)
        • Surjective and injective
        • |domain| = |codomain| = |image|
      • ...or nothing

Quiz

Let A={1,2,3,4,5} and B={a,b,c,d,e,f,g}. (∣A∣=5,∣B∣=7)

Question:

1. What is the maximum possible cardinality (size) of the Image (Range) of a function $f: A \rightarrow B$?
2. Can there exist an Injective (one-to-one) function $f: B \rightarrow A$? Explain briefly.

Answer:

1. 5. Each element in domain have mapping to codomain, so maximum size of image is equal or less than size of domain.
2. No. Every element in domain(B) must have mapping for element in codomain(A), but the size of domain is bigger than the size of codomain, so at lease 3 mapping will have same image.

• Division
  • a|b and a|c -> a|(b + c)

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