Discrete Math Examlet A

writing proof specific notes update

* make sure you use different variables * state types implicitly * state when you use definitions - and make sure definitions connect back in the end fully * show ALL steps (no logical gaps) ,, might want to revise the proof when done (includes QED) * showing number is positive when doing algebra with VIDAP (variables imlicit definition all positive)

cbtf exam taking general notes

* show all intermediate steps * ex. If i am tired or sleepy, then i will go to sleep. if finding negation, write out intermediate because you will find not tired and sleepy, and the and is important

negating quantifiers

"There exists x in a, P(x)" is false when "For all x in A, P(x) is false)" you don't negate the quantifier when finding the contrapositive

universal quantifier and existential quantifier

"for all" vs "there exists"

composition of function

(f ∘ g) (x) can be written as f[g(x)].

writing proof general concept update

* define beginning and end clearly, and go from there * think about what would not make this true, and consider proving contrapositive * analyze properties of when the claim is true or not (ex. 17 and 17 have r 0) and * proof can be in opposite order of scratch work (to fit beginning to end) * you can make claims more general (x<-5, x<0 or x | 12 then x | 36) * General things that would be useful (ex. remaindre in modular arithmetic) DNAOGG Define not analyze order general general

what numbers aren't prime

0, 1, -15 (primes must be greater than or equal to 2)

solving absolute value equations

1. Isolate absolute value 2. set up two equations 3. solve each equation

general note about solving proofs with rational study set as example

think ahead more! in this case i proved y+1/3 in a complicated manner. thinking ahead instead of just jumping into things would've allowed me to solve faster and with less steps. THINK AHEAD THINK AHEAD also read the questions closely

divisor def

: Dividend ÷ Divisor = Quotient divisor is a factor of the dividend.

propositional logic vs predicate logic

A proposition is a statement which is true or false (but never both!). predicate logic takes variables, used to make general statements

0 belongs to N but not __

Know that 0 belongs to N but not Z+.

note about use of parantheses

use to make sure order of logic is clear better to use - always

Know that sqrt(2) is not rational. how many primes are there

bruh infinite

a divides b, b is a multiple of a,

Definition: Suppose that a and b are integers. Then a divides b if b = an for some integer n. a is called a factor or divisor of b. b is called a multiple of a The shorthand for a divides b is a | b. Be careful about the order. The divisor is on the left and the multiple is on the right.

how to figure out if logical equivalence is correct or not

can use truth table or counter example, or algebraic manipulation deep though could help

Given an if/then statement, give its converse, and contrapositive. Know that the contrapositive is equivalent to the original statement, but the converse is not.

converse: p → q is q → p not equivalent (not true in all cases) contrapositive: ¬q → ¬p true

what you have to do to Simplify a negation or contrapositive by moving all negations onto individual propositions or find negation of statement

convert into logical shorthand start big, work downward make sure ALL negations are simplified

factoring a number into primes

Prime numbers, factoring a number into primes. (Where we only consider numbers >= 2.) ex. 20 = 5 * 2 * 2

definitions of classification of numbers

Remember that the rational numbers are the set of fractions of integer p and integer q p/q where q can't be zero and we consider two fractions to be the same number if they are the same when you reduce them to lowest terms. The complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the square root of -1

Know the distributive, commutative, and associative laws and that "p implies q" is equivalent to "____".

distributive (think about how addition works) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) "p implies q" is equivalent to "(not p) or q".

binding, scope, and free

domain or replacement set is where values can be taken from

Identify non-statements (e.g. questions) and statements which are neither true not false, because they contain variables not bound by a quantifier.

ex. Do you like flowers? ex. if you graph the numbers (a, b) and (c, d) on a plane, does is it always a line? is neither true or false because the numbers are not bound to a classification (they can be complex)

basic truth tables

T F is when implies is false exclusive or is false when p = t and q = t and is notated by ⊕

homework problem implication

find a counterexample here proving algebra would've worked

set membership notation

x∈Z means x belongs to the set ∈ with line crossed through it means it doesn't belong to the set

Solving inequalities check

The same as =, but reverse inequality sign if you divide or multiply by a negative number multiplying inequalities: quadratic inequalities lead to two ones that share the sign solve (x−5)(x+3)<0 (x−5)(x+3)> -3 ___ 5 x>-3 and x < 5 x + 3 > 0 and x - 5 < 0 same works for having <= if (x-5)(x+3)>0 then two solutions (x<-3 or x > 5)

Fundamental Theory of Arithmetic

a key fact of prime numbers

proof by cases

a proof broken into separate cases, where these cases cover all possibilities ex. absolute value

exponent rules

applies to reals We'll assume that √ x returns (only) the positive square root of x.

logarithm facts

importantly, notice that the multiplier to change bases is a constant, i.e doesn't depend on x. Thus, authors often write log x and don't specify the base. log b (b) = 1 log b (b^k) = k

proof by tiles

justify steps also probably get it into quadratic form and work backwards

triple equal sign usage

logcical equivalents, modular arithmetic

p->q equivalence and negation

negation: p ^ ¬q same: ¬p v q

congruent mod k

note a - b, order doesn't matter can get a = b + pos k n

translating sentence into logic

note that using and means for every x, x is a perfect square

homework - is proof correct?

note that you can make things more general (x<5 then x<=5 is true, even if the reverse isn't!)

direct number theory proof

note the different variable names

example of divides and why zero is even

notes how it applies to negative numbers

perfect square

product of an integer with itself

standard ways to approach a proof

prove a universal claim by working with a 'representative' object of the appropriate type (not a concrete example) * ex. let x be an integer prove an if/then statement by assuming whatever's in the hypothesis and proving the conclusion disprove a universal claim by giving a concrete counterexample prove an existential claim by giving specific values that make the claim true

stating the division algorithm

remainder is never negative

solving equations using i

simplify each factor as soon as possible i = sqrt -1 i ^2 = - 1 i ^ 3 = -i i ^ 4 - i

example of direct proof

start with what given end with what you need to show

a danger o multiply inequality blindly ex. (x<5) * x<=6)

then x<=30 is invalid because they can both be negative like if x<4 and y<4 xy<16 is false. you can't just multiply like that =