CSU CS220 Exam 1 prep

(recap) set 1. features 2. what are the two ways of set initiation in python 3. methods 4. set theroy operations 5. set index starts with?

1. unordered, unique, mutable 2. nums1 = set([1, 2, 3]) nums2 = { 7, 8, 9 } 3. len(aSet) set1.update(set2) # add elements in set2 to set1 aSet.add(value) set.remove(value) set.pop() # removing a random value set.clear() # remove all 4. newSet = set1.intersection(set_a, set_b, set_c...) # returning a new set of the shared items between set1 and other sets newSet = set1.union(set_a, set_b, set_c...) # all elements newSet = set1.difference(set_a, set_b, set_c...) # unique in set1 newSet = set1_a.symmetric_difference(set_2) # unique in set1 or set2 5. 1

set subtraction law

A - B = A ∩ B(Bar)

A x B will not be the same as B x A, unless

A = B, or either A or B is empty A × ∅ = ∅

function

A function f that maps elements of a set X to elements of a set Y, is a subset of X × Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x, y) ∈ f. --if not exactly one y for each x, or not all x are used, then not well-defined f: X → Y / f(x) = y

disjoint

A ∩ B = ∅

function equality

Equal if have same domain, target, and f(x) = g(x) for every element in domain

T or F: ∅ ⊂ A for any A

F, as A must != ∅

inverse

If not p, then not q ¬p → ¬q

contrapositive

If not q, then not p ¬q → ¬p

converse

If q, then p q → p

is f(x) = x^2 − 2 a complete definition

No, range and target should be specified

(recap) types for collections in python

Sequence types string, list, and tuple set type for set The only mapping type in Python is the dict type

T of F: in a set, element type can be different

T

set intersection

The intersection of two sets is the set of all the elements that are common to both sets. The intersection of A and B is written as A⋂B and includes those elements that are in set A and in set B. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋂B = {a, e}

set union

The union of two sets is the set of every element that is in either or both sets. The union of sets A and B is written as A⋃B and includes those elements in either set A, set B, or both. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋃B = {a, b, c, d, e, i, o, u}

function composition

Using the output from one function call as the input to another. f and g are two functions, where f: X → Y and g: Y → Z. The composition of g with f, denoted g ο f, is the function (g ο f): X → Z, such that for all x ∈ X, (g ο f)(x) = g(f(x))

geometric sequence

a sequence in which each term is found by multiplying the previous term by the same number a, ar,ar2,...,arn,...

Cartesian product

a set of pairs (x, y) of elements from two sets X and Y. A x B = { (a, b) : a ∈ A and b ∈ B } If A = {1, 2, 3}, then A × ∅ = ∅

sequence

a special type of function in which domain is a consecutive set of integers

range

actual outputs, subset of Y

proposition

an argument that is logically true of false (or unknown) --in English, typically declarative sentences, can't be questions or commands

arithmetic sequences

an=a1+(n-1)d

conjunction

and, ∧

exponential

b is base and b != 1, x is exponent

operator precedence

calculation order: quantifiers, not, and, or, → , ↔

set builder notation

characterizing by the property its elements satisfy ex., O = {x: x is an odd positive interger less than 100}

domain

elements in set X, inputs

biconditional

if and only if, p ↔ q return true if p = q

contradiction

if the proposition is always false, regardless of the truth value of the individual propositions that occur in it

tautology

if the proposition is always true, regardless of the truth value of the individual propositions that occur in it

bijective

injective and surjective

log example, starting with n chocolates, dividing the chocolotes into two halves and keep one half, the times k needed to reach 1 chocolate left is ?

log2n if keep the larger half, then ceiling(log2n), if keep the smaller half, then floor(log2n)

inverse of exponential is called

logarithm

floor function

maps a real number to the nearest integer downwards

ceiling function

maps a real number to the nearest integer upwards

contingency

neither tautology nor contradiction

set complement

not in A but in U, = U - A

injective

one-to-one, one y mapping only one x In proof: if y1 = y2, then x1 = x2

comparing the size of domain and target for one-to-one, onto, and bijective

one-to-one: domain <= target onto: domain >= target bijective: domain = target

inverse of a function

only if f: X → Y is a bijection, then f-1 = { (y, x) : (x, y) ∈ f } how to get f-1(x): let y = f(x), switch y and x, find y

surjective

onto, range = target, all target elements are used In proof: for every y ∈ Y , there is a (well-defined) x = sth (figure out x) ∈ X such that f(x) = y --prove Y, prove x = sth, prove X, prove f(x) = y

disjunction

or (inclusive) --in English, ppl often use exclusive or, p ⊕ q

identity function

output = input IA : A → A, IA(a) = a Let f: A → B be a bijection. Then f-1 ο f = IA and f ο f-1 = IB --IA is starting from A

conditional proposition / conditionals

p → q, p implies q, p only if q, the proposition p is called the hypothesis, and the proposition q is called the conclusion --think of the circumstance of p = true and q = true as a contract, the only way to break the contract is when p = true and q = false p → q ≡ ¬p ∨ q

set partition

partitioning of elements of some universal set A into a collection of disjointed subsets. Thus, each element must be in exactly one subset. For all i, Ai ⊆ A. For all i, Ai ≠ ∅ A1, A2, ...,An are pairwise disjoint. A = A1 ∪ A2 ∪ ... ∪ An

logarithm

properties: logb(xy) = logbx + logby logb(x/y) = logbx - logby logb(x^y) = ylogbx logcx = logbx / logbc

target

set Y, possible outputs

set identity

set caculations / equations involving sets that are true for ALL sets

logically equivalent

statements with the same truth values, regardless of the truth values of their individual propositions p ≡ q

roster notation

the elements of a set are listed between braces

cardinality

the number of elements in a set, |aSet|

power set

the set of all subsets of a set ex. A = { 1, 2, 3 }, then: P(A) = { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } } remember to include the ∅ in the front! P(ø) = {ø}

set difference

unique in A A - B, what's left in A after substracting the intersection subtraction is not associative

symmetric difference

unique in A + unique in B A ⊕ B = (A-B) U (B-A) = (A U B) - (A ⋂ B) symmetric difference operation is associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)

strictly increasing

whenever x1 < x2, then f(x1) < f(x2)

name x and y: A binary string is a string whose alphabet is {x, y}

x = 0 y = 1

negation

¬p --translate English negation: "It's not true that..."

The empty string is usually denoted by the symbol __

λ

∧, ∨, ¬

∧: and, ∨: or, ¬: not