~CHAPTER 0 .... Preface to the students........Study of mathematical reasoning

Goal, instead of calculating/solving like in Calculus etc is:...advanced courses are more about establish certain mathematical structure which has certain properties.

THIS COURSE bridges those two goals.

DEDUCTIVE REASONING: Mathematicians use logic to develop/extend a theory, drawing conclusions based on statements they accept as true.

INDUCTIVE REASONING: Used to develop a theory.

PROOFS: Demonstrate the truth in the conclusions that are drawn.

TRYING A WHOLE BUNCH OF NUMBERS DOES NOT WORK, WE NEED PROOF THAT IT WORKS FOR ALL NUMBERS!!!

We will learn techniques of reasoning and proof.

The FIRST GOAL is to learn standard techniques for developing proofs....especially how to get started on a proof, and how to construct proofs using those techniques.

We'll pull from the logical form of a statement and use it as a guide for constructing a proof.

From this we will learn of the ability to reason ABSTRACTLY, which helps in advanced math and beyond.

TO KNOW BEFORE BEGINNING CHAPTER 1:

-Calculus-"If and only if" is very important-A Set is a collection of Elements-Elements are members of a set (objects)

means "belongs to..." eg

...means "x is an element in set A"

eg: A is a subset of B IFF every element of A is an element of B.

If A and B have the same exact elements, then A = B.

Sets are finite of they have nothing, or has "n" elements.Otherwise, sets are infinite.

NATURAL NUMBER PROPERTIES

1. Successor (Think of ONE): *no integers a.1 is a natural number. b. Every natural number has UNIQUE successor....x + 1 c. 1 is not the successor of ANY NUMBER

2. Closure (natural + natural = natural....natural * natural = natural):*integers are cool.If you add two naturals numbers, the result is naturalIf you multiply two natural numbers, the result is natural. (What if you square, cube etc? I guess that may be the same as multiplying)

3. Associativity (if added or multiplied you can move the parenthesis): NOTE: Gotta be natural numbers....integers are cool.

a. If all are added, the parenthesis can encapsulate either eg (x + y) + z = x + (y + z) b. Same with multiplying x(yz)= (xy)z

4. Commutativity (You can rearrange the numbers in a different order):NOTE: gotta be natural numbers....integers are cool.

a. x + y = y + x b. xy = yx

5. Distributivity (If there are two of the same...make it one and move it outside of the parentheses):NOTE: gotta be natural numbers....integers are cool. a. xy + xz = x(y+z)

6. Cancellation (if x plus something equals y plus the same something, x equals y. Same for multiplication)NOTE: gotta be natural numbers....integers are cool...just NO ZERO!

a. x + z = y + z.....then x = y b. xz = yz....x= y

Dividing thing:...integers are cool...

a divides b if there's a "k" such that b =ak.

eg: 9 and 90

9 divides 90 because there's a number 10 such that 90 = 9*10

PRIME: Bigger than 1 and only 1 and itself divide into it.

COMPOSITE: ain't 1 nor is it a prime

If a natural number is bigger than 1 it is a prime or product of primes.

EVEN INTEGER: Iff there's a k such that 2k = the number.

ODD Integer: Iff there's a 'j' such that 2j + 1 = the number.

RATIONAL NUMBER: x is rational if x = p/q..... and q ain't zero.

Rationals on number line if they have TERMINAL or REPEATING decimal expressions!

ALL OTHER REAL NUMBERS, including Are IRRATIONAL

REAL AND RATIONAL NUMBERS HAVE A "MULTIPLICATIVE INVERSE" eg there's a 'y' such that xy = 1

Duh! That just means reciprocal!!!

Complex Numbers: a + bi.....and i is

The CONJUGATE of a + bi is a - bi. REAL NUMBERS are a subset of COMPLEX NUMBERS, as any real number can be x+0i. ...different ordering properties however.

FUNCTIONS: (mapping)

Unique element of b is assigned to each element in a. Can be numbers, functions, or vaginas.

"f is a function from A to B":

If a A, then f(a) = b.

The elements of a are called arguments or inputs of the function.

If f(a)=b then a is the "pre-image" of b.....and b is the "image of a".

f(pre-image)=image

Preimage really means the number that gets put through the f wringer. Image is what the preimage becomes after it has been put through the wringer.

SINGLE VALUED PROPERTY: A can only have UNIQUE images of elements, and an image in A exists in B!

DOMAINS AND CODOMAINS:

"f maps A to B"

SECTION 1.2

"If....then"

THIS CONDITIONAL IS TRUE IFF:P is false, or Q is true

CONVERSE

CONTRAPOSITIVE

BICONDITIONAL

THIS MEANS THAT FOR THE TRUTH TABLES OF P AND Q .... P<=>Q is only true for TWO VALUES THAT A THE SAME .... LIKE AN XNOR GATE!!!

1.4