Answer as many questions as you can.

1. Write (in the logic notation used in this class) an encoding of “Everyone in Tanya’s class is more likely to pass the midterm if they know Type Theory”. Context: Tanya is a student and is only taking one class right now. “Everyone” means Tanya and her classmates.
2. A logical system has three parts. Choose the best description for the three parts:
   $\bigcirc$ syntax, axioms, inference rules

   $\bigcirc$ axioms, inference rules, theorems

   $\bigcirc$ alphabet, grammar, inference rules

   $\bigcirc$ symbols, models, inference rules

   $\bigcirc$ syntax, semantics, inference rules

   $\bigcirc$ symbols, axioms, models
3. Which of the following are accurate descriptions of 5 pentated to 2?
   $\Box$ 25

   $\Box$ A power tower of five 2s

   $\Box$ A number that would take weeks to write out by hand

   $\Box$ $5\uparrow\uparrow 5\uparrow\uparrow 5\uparrow\uparrow 5\uparrow\uparrow 5$

   $\Box$ $5^{5^{5^{5^5}}}$
4. Let $R = \{ (1,2), (2,1), (3,3) \}$. What is $R^2$ assuming (a) $R$ is just a regular set, and (b) $R$ is a relation?
5. Write the cond function in lambda notation.
6. Match the people on the left with something they are known for on the right.

     Bertrand Russell          Entscheidungsproblem
     Abraham Fränkel           Replacement
     Ernst Zermelo             Lambda Calculus
     David Hilbert             A letter to Frege
     Ada Lovelace              Bernoulli Numbers Program
     Alonzo Church             Infinities
     Georg Cantor              Axiomatizing Set Theory
   

7. Match the Set Theory axioms with their intent.

     Foundation                You can combine two sets
     Extensionality            Set of all subsets is a set
     Pairing                   The natural numbers comprise a set
     Union                     You can make a set by filtering
     Power Set                 No self-containing sets
     Replacement               Set of all members' images under a function
     Infinity                  All the elements of the elements are a set
     Choice                    Set with one element from each member
     Separation                What equality means
   

8. How is the list $[1,2,3]$ encoded in ZFC?
9. Give an inductive definition for a type which we will call Propositional Logic Formulas, or $\textsf{PLF}$. PLF’s are either constants, variables, not-expressions, and-expressions, or-expressions, or material implication expressions. You should be able to take it from here.
10. What are the inhabitants of the type $\textsf{Bool}\to\textsf{Bool}$? Write them out precisely.