CSCI 301 - Midterm Exam Study Guide

Exam Details

• In class on Monday, May 5th
• Covers material through Wednesday, April 30th (through BoP Chapter 12; up to and including functions).
• You can bring one double-sided 8.5x11” sheet of hand-written notes.
• Resources:
  • (you are here) I have made a study guide of sorts, which lists of the Goals for each lecture; my goal is for the exam to be a fair assessment of the extent to which you’ve achieved these outcomes.
  • I have provided some Racket practice problems. You can expect the Racket questions on the exam to resemble these.
  • These are also linked from the course webpage Schedule table, on the day of the Midterm Exam.

For practice problems, see the Racket practice problems, and the in-class exercises (linked as ws from each day’s lecture on the Schedule table). See also the exercises for BoP Chapters 1-2 and 4-10; odd numbered exercise solutions are in the back of the book.

Course Outcomes:

The course outcomes relevant to this exam are the following, with some parts crossed out that we haven’t covered yet:

1. Thorough understanding of the mathematical definitions of concepts important to computer science, including sets, tuples, lists, strings, formal languages, functions and relations.
2. The ability to prove basic theorems involving these mathematical concepts including proofs by contradiction and inductive proof
3. The ability to employ effectively the functional programming style in a functional programming language

Per-Lecture Goals

Racket 1

• Know how to interpret and write expressions in prefix operator notation

• Know how to interact with the Racket interpeter in DrRacket

• Know how to represent and work with basic data types (booleans, numbers, strings) in racket.

• Know how to define and use functions in Racket using lambda

• Know how lexical scope works in Racket, and how to introduce local variables using let and let*.

Racket 2

• Know how to express conditionals using if and cond
• Know how to create and work with dotted pairs using cons, car, cdr, and cadr and friends
• Know how create and interact with lists, including the list function
• Be able to implement simple recursive list processing functions in Racket
• Know how to create recursive helper methods to carry information through the recursion

Sets 1

• Describe what a set is and how a set’s contents and members are denoted (\(\{\ldots\}\),\(\in\) )

• Know the notation for natural numbers (\(\mathbb{N}\)), integers (\(\mathbb{Z}\)), rational numbers (\(\mathbb{Q}\)), real numbers, and the empty set (\({\varnothing}\) or \(\emptyset\), or occasionally \(\{\}\)).

• Define sets using set-builder notation.

• Calculate the cardinality of a set.

Sets 2

• Know the definition of subset, power set.
• Compute a Cartesian Product of two or more sets.
• Understand the notation and meaning for: union, intersection, difference, and complement
• Be able to draw and interpret Venn diagrams

Logic 1

• determine whether an English sentence or a mathematical expression is a logical statement.
• articulate the meanings of and know the truth table for logical AND, OR, NOT, operators
• articulate the meaning of and know the truth table for the conditional operator

Logic 2

• settle the debate over “only if” once and for all!
• articulate the meaning of and know the truth table for the biconditional operator
• use truth tables to calculate truth values of logical statements
• be able to interpret and use universal \((\forall)\) and existential \((\exists)\) quantifiers
• Be able to translate English sentences involving quantifiers, logical operators, and statements into logical symbols and back.

Logic 3

• know (or know of the existence of so you can later look up) some useful equivalences, including the contrapositive, de Morgan’s laws, and negation of quantifiers and conditionals
• Be able to negate statements expressed symbolically.

Proofs - Direct Proof

• Know how to read a mathematical definition, and know a few common and useful ones:
  • even, odd, divides, parity
• Be able to use direct proof to prove statements involving definitions including even, odd, divides, parity.

Contrapositive Proof

• Know how to use contrapositive proof to show \(P \Rightarrow Q\) by showing that \(\neg Q \Rightarrow \neg P\).
• Practice writing proofs.

Modular Equivalence

• Know and be able to apply the definition of congruence modulo \(\mathbf{n}\)

Proof By Contradiction

• Know how to prove statements, including conditional statements, by contradiction.

Proving Non-Conditional Statements

• Know how to prove if-and-only-if statements and equivalence statements
• Know how to prove existentially quantified statements
• Know how to disprove statements in general
• Know how to disprove universally quantified statements and conditional statements by counterexample

Proofs involving Sets

• Know how to prove \(a \in \{x: P(x)\}\) and \(a \in \{x \in S: P(x)\}\)
• Know how to prove \(A \subseteq B\).
• Know how to prove \(A = B\) for two sets \(A, B\).

Proof by Induction 1

• Know the intuition and logical reasining underlying proof by induction.
• Know the meaning of base case, inductive hypothesis, and inductive step (or inductive case) in the context of proof by induction.
• Be able to apply proof by induction

Proof By Induction 2

• Know the definition of the Fibonacci sequence
• Be able to apply proof by strong induction
• Get more practice doing inductive proofs

Relations

• Know the definition of a relation on a set.
• Be able to apply the definitions of the following properties of relations:
  • Reflexive
  • Symmetric
  • Transitive