Math 365

• 1. Understand the problem
• 2. Devise a plan
• 3. Carry out the plan
• 4. Looking back

Can you state the problem in your own words

Look for a pattern

Check each step of the plan as you proceed

check the results in the original problem

a statement throught to be true, but not proven

example that contradicts the conjecture, shows the conjecture false

a_n= a₁+ d(n-1)

a_n = a₁* r^(n-1)

Ex: a₁=2, a₂=3, a_n=3a_n-2-a_n-1, for natural #n>2

must have all 3 parts or will be wrong

either T or F

• Be careful w quantifiers:
• Universal: all, every, & no refers to each & every element in a set

Existential: some, there exists at least one refers to one or more or passible all elements in a set

• p^q (p and q) - if both are T then its T
• pVq (p or q) - if both are F then its F

• Conditional Statements:
• p --> q (if p then q)
• Converse:
• q --> p (if q then p)
• Inverse:
• ~p --> ~q (if not p then not q)
• Contrapositive:
• ~q --> ~p (if not q them not p)
• Biconditional:
• p <--> q (p iff q)

*If 1st is T & 2nd is F then its F*

assigns a value of a digit depending upon its placement in a numeral

if a is any # and n e N, then aⁿ= a*a*...*a

Ex: 2³= 2*2*2=8

• a⁰=1
• a¹=20
• a²=20*18=360
• a³=20²*18=7200...etc

• gross = dozen dozen
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E

if elements of P and Q can be paired so that for each element of P there is exactly one element of Q, & for each element of Q there is exactly one element of P

If event M can occur in m ways, and after it has occurred, event N can occur in n ways, then event M followed by event N can occure in mn ways

iff there exists a 1-1 correspondence btwn the two sets.

indicates the # of elelments in set A

if its cardinal number is a whole #

is the set of all elements in the universal set U that are not in A

• for any set A, either {}c A, or {} c A. Suppose{}c A, then there is some element in the empty set that is not in A, but because {} has no elements, it cannot have an element that is not in A.
• therefore {}c A

are an application of set concepts

If A and B are finite sets then n(A) is less than n(B), written n(A)<n(B), if A is equicalent to a proper subset of B. So if n(A)=a & n(B)=b, then a<b. Similarly we define greater than: n(A)>n(B) or a>b, which is n(B)<n(A) or b<a, respectively.

it has 2^n(A)subsets

it has 2^n(A)-1

meaning in B but not in A

Let A and B be disjoint ^{(A intercect B=0)} finite sets: If n(A)=a and n(B)=b, then a+b=n(A u B)

for any a,b e W, a is less than b, written a<b, iff there exists a k e N such that a+k=b

• Closure: if m,n e W, then m+n e W;
• Commutative: a+b = b+a
• Associative: (a+b)+c = a+(b+c)
• Unique Identity 0: a+0=0+a=a

for any a, b e W, such that a > b, a-b is a unique c eW such that a=b+c

• Start at zero facing the (+) direction
• Add means stay facing same direction
• Subtact means turn around
• (+) # means go forward
• (-) # means go backwards

• 125
• 345
• + 79
• 19 add ones
• 130 add tens
• +400 add hundreds
• 549

• 458
• +832
• 1200 (400+800)
• 80 (50+30)
• + 10 (8+2)
• 1200
• + 90 (80+10)
• 1290