CSCI 301 L02 Notes

Lecture 2 - Notes

Note: italicized outcomes are the ones we didn’t get to during Lecture 2.

Outcomes:

Racket 1

• 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*.
• Know how to express conditionals using if and cond

Sets 2a

• Know the definition of subset, power set.

Announcements

• A1 is out. You’ll write your answers in LaTeX.
  • See this guide to get started. This is also now linked from the Assignments section of the syllabus.
• Lab 1 is tomorrow
• I made a Discord server - link is on Canvas.
  • Please note: server nickname must be your real name, avatar must be SFW, and you must be a good human.
  • Contributions welcome: channel suggestions, server image, etc.

Outline

Racket 1

Defining global variables

(define base 4) ; syntax: (define <symbol> <value>)
(define height 2)
base

; area of a triangle with base 4 and height 2 = 1/2 * base * height
(/ (* base height) 2)

Functions (or procedures) are created with the lambda special form:

(lambda (x) (+ x 2)) ; evaluates to a function (or procedure)
((lambda (x) (+ x 2)) 4) ; applies the function to the argument 4

(define addxy (lambda (x y) (+ x  y))) ; store a function in a global variable
(addxy 4 8) ; apply the function to 2 arguments

Do exercise Part A

Lexical Scope

Scope in Racket is lexical - variables are visible only within a certain segment of the program text. define creates global variables whose scope is the entire program.

Function parameters (x and y, in the addxy example above) are local variables. They’re only visible inside the procedure body, and they shadow any global variable with the same name.

(define x 1)
(addxy 4 5) ; should be 9, because the x inside addxy gets bound to 4, shadowing global x

If you want to create local variables that are not function parameters, you can do that with let:

(let ((var1 val1)
      (var2 val2)
      (var3 val3))
  (expression-using-variables))
  
; var1 and friends are not visible here

(define a 4)

(let ((a 1) ; shadows the global a
      (b 2))
  (+ a b))

The local variables initializations happen (conceptually, though not actually) in parallel. This means a later one can’t reference an earlier one:

(let ((a 1)
      (b (+ a 1))) ; error: a is undefined!
  (+ a b)) 

We could nest lets to make this work:

(let ((a 1))
  (let ((b (+ a 1)))
    (+ a b))) ; this works

There’s a shorthand syntax for this - let*:

(let* ((a 1)
       (b (+ a 1)))
  (+ a b))

Conditionals

(if test val-if-true val-if-false)

(define absolute-value ; abs is a builtin
  (lambda (x)
    (if (< x 0)
        (- x)
        x)))

Equivalent to your familiar if/elseif/elseif/else chain:

(cond ((test1 val-if-test1)
       (test2 val-if-test2)
       (else val-otherwise)))

(define sign
  (lambda (x)
    (cond ((< x 0) -1)
          ((= x 0) 0)
          (else 1))))

Do Exercises Part B - I recommend working through these at home tonight if we don’t have time to finish them in class.

(End of what we covered in Lecture 1)

Sets 2a

Subsets and Powersets

Definition: Suppose \(A\) and \(B\) are sets. If every element of \(A\) is also an element of \(B\), then \(A\) is a subset of \(B\); this is denoted \(A \subseteq B\).

For example, let:

• \(A = \{1, 2, 3\}\)
• \(B = \{2, 3, 1\}\)
• \(C = \{1, 2\}\)

Then the following are true:

1. \(A \subseteq B\)
2. \(B \subseteq A\)
3. \(C \subseteq A\)
4. \(C \not\in A\)
5. \(A \not\subseteq C\)
6. \(\{n^2 : n \in \mathbb{Z}\} \subseteq \mathbb{Z}\)

Aside: If \(A \subseteq B\) and \(A \ne B\), then \(A\) is a proper subset of \(B\), denoted \(A \subset B\).

Definition: The power set of a set \(A\) is the set of all subsets of \(A\). It is denoted \(\mathcal{P}(A)\), or sometimes \(2^A\). In other words, \(\mathcal{P}(A) = \{S : S \subseteq A\}\).

Do Exercises Part C