Functions as relations

• Read carefully pages 216 through 219 (sections 9.1, 9.2)
• Some key questions to answer:
  1. A function from a set \(A\) to \(B\) is a special kind of relation from \(A\) to \(B\). What special property does it need to satisfy?
  2. What other name do we use for a function?
  3. How do we call the sets \(A\) and \(B\) in relation to a function \(f\colon A\to B\)?
  4. Explain why when we think of a function \(f\) as a relation, i.e. a subset of \(A\times B\), then the cardinality (number of elements) of that relation must equal the cardinality of \(A\).
  5. What do we define as the range of a function \(f\)? How does it differ from the codomain?
  6. Consider the empty relation. When can that relation be a function?
  7. Consider the full relation \(A\times B\). When can that relation be a function?
  8. When do we say that two functions are equal? Prove that this condition is equivalent to asking for the two functions to be equal as subsets of \(A\times B\).
  9. Think about the following sentence: “If we have a real function \(y=f(x)\) from Calculus, then its graph, a subset of \(\mathbb{R}\times\mathbb{R}\) is exactly what this function is thought of as a relation”.
  10. How many functions are there from the set \(A=\{1,2,3\}\) to the set \(B=\{x, y\}\)?
  11. In general, how many functions are there from a set with \(k\) elements to a set with \(n\) elements?
  12. Given a function \(f\colon A\to B\) and a subset \(C\) of \(A\), what do we denote as the image set \(f(C)\)?
  13. Given a function \(f\colon A\to B\) and a subset \(D\) of \(B\), what do we denote as the inverse image set \(f^{-1}(D)\)? Note that this is defined regardless of whether the function \(f\) has an “inverse” in the sense learned in Calculus.
  14. What is \(f(\emptyset)\)? What is \(f^{-1}(\emptyset)\)?
  15. True or False: \(x\in f^{-1}(D)\) if and only if \(f(x) \in D\).
  16. Food for thought: The set of all functions \(f\colon A\to\{0,1\}\) is in correspondence with the subsets of \(A\):
      • Given such a function, we can associate it to the set \(B=f^{-1}(\{1\})\subseteq A\).
      • Given a subset \(B\subseteq A\), we can build a function \(f\colon A\to\{0,1\}\) so that \(f(x) = 1\) if \(x\in B\) and \(f(x) = 0\) if \(x\not\in B\).
      • This is often called the “incidence function” or “characteristic function” of the subset.
  17. Suppose \(f\colon A\to B\) is a function, thought of as a relation, i.e. subset of \(A\times B\). Suppose \(C\subseteq A\) is a subset of \(A\). Consider the relation \(f \cap \left(C\times B\right)\). Show that this defines a function from \(C\) to \(B\). This is known as the restriction of the domain of \(f\) to \(C\).
• Practice problems from section 9.1 (page 234): 9.1, 9.3, 9.5, 9.7, 9.8, 9.9, 9.11, 9.12
• Practice problems from section 9.2 (page 235): 9.15, 9.16