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91 Cards in this Set
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What is Deductive Reasoning?

Given true statements, we can make an argument that another statement is also true.
The true statement we start with is the hypothesis, or premises. The true statement we end up at is the conclusion. 

What is the Rule of Direct Reasoning?

Given p>q is true, and p is true, we know that q must be true.
**If p>q is true and q is true, we can not infer anything about p .: this is invalid direct reasoning** 

What is the Rule of Indirect reasoning?

Given p>q is true and q is false, then we know that p must be false.
**If p>q is true and p is false, we can not infer anything about q .: it is invalid indirect reasoning** 

How do you make a proof by contradiction?

Given p>q, assume that q (or p) is false, while the other remains true. If the conclusion reached is also false, then you have proved p>q to be true.


What is a set?

A collection of objects or ideas that can be listed or described
we use capital letters to denote a set (set A) 

What is Universe?

Universe (U) is the bigger group that a set is derived from.
Ex. the primary colors is a set of all colors, where all colors is U 

What are the three ways to define a set?

1) words
2) list in {} 3) set builder notation 

What is set builder notation?

Ex: {x ∊ U  7<x<12}
(x is in U such that 7 is less than x which is less than 12) {8,9,10,11} 

what does x ∊ S mean?

that x belongs to set S


What is a Venn Diagram?

A rectangle with a circle (or more than one circle) in it.
The area inside the circle(s) represents the set(s) The area outside the circle(s) represents the Universe 

What is compliment?

Compliment of set A (Ā) is the set of elements in U that aren't in A


What is subset?

Set A is a subset of set B when all of A is in B. Denoted A ⊆ B


What is a Proper Subest?

Set A is a proper subset of set B when all of A is in B, but not all of B is in A. Denoted A ⊂ B


What is intersection?

When two sets overlap, the elements that are in both is the intersection. denoted A ⋂ B


What is union?

Everything that is in two sets, whether it overlaps or not. Denoted A ⋃ B


What is disjoint?

When two sets are completely separate, with no overlap. Also known as Musually Exclusive.


What is transtivity of inclusion in set operations?

If A⊆B and B⊆C then A⊆C


What is communativity in set operations?

AUB=BUA and A⋂B=B⋂A


What is associativity in set operations?

AU(BUC)=(AUB)UC and A⋂(B⋂C)=(A⋂B)⋂C


What is AU∅

A


What is A⋂∅

∅


What are the distributive properties in set operations?

A⋂(BUC)=(A⋂B)U(A⋂C) and AU(B⋂C)=(AUB)⋂(AUC)


What is a nominal number?

A number used as a name or a label (such as a bank number, or a seat number on a ticket)


What is an ordinal number?

A number used to describe the relative position of an object in an ordered sequence (such as the 14th row, or second, or July 13)


What is a cardinal number?

The number of objects in a set (such as 5 items on the shopping list)


What is a onetoone correspondance?

When two sets, A and B, have the same amount of elemnts. A={red, blue, yellow} B={cat, dog, frog} A and B have a onetoone correspondance.


What are equivalent sets?

Two sets A, B, are equivalent if they have a onetoone correspondance. Denoted A~B


What is a finite set?

A set with a limited number of elements


What is an inifinite set?

A set with an unlimited number of elements, such as the set of whole numbers


What is the cardinality of set A?

The number of elements in set A, denoted n(A)


What's the cardinality of the empty set?

n(∅)=0


What are whole numbers?

The cardinal numbers of finite sets


What are three ways to represent whole numbers?

Tiles, number strips (cuisenaire rods), and a number line


How do we define numbers on the number line

3 is a distance of 3 from zero


How would we use sets to show that 3 is less than 5?

For whole numbers a and b, sets A and B with n(A)=a and n(B)=b, a is less than b (a<b) if A is a proper subset of B


How do we use a number line to show that 3 is less than 5?

draw a number line, and use an arrow to go to 3, and to 5...the arrow to 5 will be bigger, and longer, :. 3<5


Define addition of whole numbers, using sets

if A and B are disjoint sets, n(A)=a and n(B)=b, then n(AUB)=a+b


Define the parts in an addition equation

a+b a and b are the addends, a+b is teh sum of a and b


what is the equation for n(AUB)?

n(AUB)=n(A)+n(AB)n(A⋂B)
you would use this equation when sets are not disjoint 

How would you use a number line to do addition?

Start at zero, and go a distance of the first number. The second number's arrow tail starts at the head of the first number's arrow. (If you were adding 3+4, you would start at zero and go a distance of 3. then start at 3 and go a distance of 4 for a total of 7)


What is the Closure Property in addition of whole numbers?

For whole numbers a and b the sum a+b is a unique number. This means that when any two numbers in a set are added, the result is in the same set (ie the set of whole numbers is closed because it is an infinite set)


What is the Identity Property in addition of whole numbers?

There exists a unique number, 0, such that 0+a=a for every whole number a. Zero is the additive identity element.


What is the Commutative Property in addition?

For whole numbers a and b a+b=b+a


What is the Associative Property in addition?

For whole numbers a,b,c a+(b+c)=(a+b)+c


Define subtraction of whole numbers

For whole numbers a and b, the difference ab is the unique number c, such that a=b+c


Define the parts in subtraction of whole numbers

In ab=c a is the minaend, b is the subtrahend, c is the difference of a and b


What are the four conceptual models for subtraction?

Takeaway, Missing Addend, Comparison, and Number Line


What are the steps for the TakeAway model?

1)Start with x number of objects
2)Take away y number of objects 3)How many are left over? (z objects) 

What are the steps for the Missing Addend model?

1)Start with y objects
2)How many more objects are needed to get x objects? (z objects) 

What are the steps for the Comparison Model?

1)Start with 2 collections, x and y
2)How many more x do I have than y? (z objects) 

What are the steps for the Number Line Model?

1)Move forward (to the right) x units
2)Remove a jump to the right of y units (coount back from the head of the first arrow y units. place the tail of you arrow here, and the head is the same as the head of the first arrow) 3)What is the distance from 0? (z units) (what's the distance from 0 to the tail of the second arrow) 

Does subtraction have the Closure Property?

No. For example, in whole numbers 25 gives me an undefined number; there is no whole number that gives me 2=5+a


Does subtraction have the commutative property?

No. 52=3
25=undefined :. 52≠25 

Does subtraction have the associative property?

No. a(bc)≠(ab)c
Ex: a=7, b=3, c=2 a(bc)=7(32)=71=6 (ab)c=(73)2=42=4 6≠4 

how is multiplication defined by repition?

Given two whole numbers a, b, the product of a and b (a⋅b) is defined by a⋅b=b+b+...+b (a number of b's) such that a≠o


how is multiplaction denoted?

a⋅b, axb, a*b, ab, (a)(b)


what are the parts of the equation a⋅b

a and b are factors; a⋅b is the product


What are the four models of multiplication?

Array, rectangular area, cartesian product, and tree diagram


What is an array?

An array uses a diagram of objects (such as four blocks high by four blocks wide to illustrate 4⋅4). It could also just be groups of objects.


What is rectangular area model?

The two numbers beings multiplied represent the dimensions of a rectangle. The area is the result of the multiplication.


What is an ordered pair?

2 entries (from two different sets) between brackets and separated by a comma. Ex. (cat, dog) which is different from (dog, cat)


What is a cartesian product?

It involves 2 sets A and B, denoted AxB. It is the set of all ordered pairs (x,y) such that x is an element of A, y is an element of B.
Also denoted AxB={(x,y)x∊A and y∊B} The number of ordered pairs is equal to n(A)⋅n(B) 

What is a tree diagram?

It uses "branches" to denote the different possibilites. Example, if you have three types of cookies, and 2 types of drinks, you'd start with 2 branches  cookies and drinks. Then you'd have 3 branches from cookies for the different types, and 2 branches for drinks. The total number of end branches is the answer.


What is closure in multiplication?

The product of two whole numbers, a,b, a⋅b is a unique whole number.


What is the multiplicitave identity of one?

1 is a unique whole number such that b⋅1 and 1⋅b will always equal b for any given whole number b.


What is the multiplication by 0 property?

For all whole numbers b, b⋅0 = 0⋅b = 0


What is commutativity in multiplication?

For any two whole numbers a and b a⋅b=b⋅a


What is associativity in multiplication?

For any three whole numbers a, b, c a⋅(b⋅c) = (a⋅b)⋅c


What is the distributive property in multiplication?

Given three whole numbers a, b, c a⋅(b+c) = (a⋅b)+(a⋅c) and a⋅(bc) = (a⋅b)(a⋅c)


How is division defined?

Division is the inverse of multiplication. In multiplication you multiply two factors together to get a product. In division, you divide the product by a factor to get the other factor.
Given 2 whole numbers a, b, and b≠0 then a/b=c is true if c is a unique whole number such that a=bxc 

What are the parts of the equation a/b=c

a is the dividend, b is the divisor and c is the quotient


What are the three models of deivision?

partitioning, repeated subtraction, and the division algorithm.


What is Repeated Subtraction?

In multiplication we have repeated addition, where axb=b+b+...+b a times
In division, a/b = abb...b where the number of times b is subtracted equals c, the quotient 

What is partitioning?

separate a set into a known number of equivalent subsets and ask "how many elements did I put into each subset?"
IE. 300 invitations, 25 rubber bands..how many invitations in each rubber band? 

What is the division algorithm?

Given whole numbers, a and b, such that b≠0, then there is a unique whole number q, the quotient, and a unique whole number r, the remainder such that
a=b⋅q+r OR a/b=qRr when 0≤rb q is the largest number of b's that you can subtract from a, and r is the Remainder  the number of objects left over 

Why is division by zero undefined?

ex. 8/0 has no solution because 8/0 must be the same as 8=0c. There is no number you can multiply 0 by to get 8.
ex. 0/0 has multiple solutions. 0/0 is the same as 0=0c. There is an unlimited amount of numbers that you can multiply 0 by to get 0. However, c must be a unique number. 

Can we have 0 as the dividend?

yes. Ex. 0/8 is the same as 0=8⋅c. 8⋅0=0 :. 0/8=0. 0 divided by any number equals 0.


What properties does division have?

Division does not have:
closure, commutative, associative, distributive properties. 

What is an exponential number?

Given two whole numbers, a and m, such that m≠0, then a^m can be defined as a⋅a⋅a⋅...⋅a m times.


What is the rule for a^m if m=0? m=1?

if m=0, then a^m=1
if m=1, then a^m=a 

What are the three rules for multiplying exponential numbers?

Given whole numbers a,b,m,n such that m,n≠0
1. a^m⋅a^n=a^(m+n) 2. a^m⋅a^n=(a*b)^m 3. (a^m)^n=a^(m*n) 

What are the two rules for dividing exponential numbers?

Let a,b,m,n be whole numbers such that a,n≠0
1. a^m/a^n=a^(mn) 2. a^m/b^m=(a/b)^m 

What is a numeration system?

symbols for writing numbers (numerals) together with methods for calculation (algorithms)


What is place value?

the position or location of a numeral to determine its value


What are the characteristics of the Egyptian system?

used pictures (hieroglypics) to represnt powers of 10 (it was a base 10 system)
it had no symbol for zero they had no place value 

What are the characteristics of the Babylonian system?

introduced the notion of place value
based on multiples of 60 had 2 symbols combined the symbols additively to form 159, after 60 they used the symbols to write powers of 60 place values were units, 60, 60^2, 60^3, etc. Originally didn't have a symbol for zero, but introduced it later on 

What are the characteristics of the Mayan system?

it was based on their calendar
Used a modified base 20 Included a symbol for zero Units were 119, after that it used the modified place value: units, 20, 18⋅20, 18⋅20^2, etc. Their numbers were displayed vertically, with units at the bottom Symbol for zero 

What are the characteristics of the Roman system?

Used a mix of base 10 and base 5
Seven basic symbols: I=1, V=5, X=10, L=50, C=100, D=500, M=1000 Combined and repeated symbols as necessary Could only repeat a symbol 3 times, so to get the 4th, they would use that that symbol before the symbol of one higher, meaning "one less than" (it. ab=a less than b..IV for 4, CM for 900) Very large numbers are represented by a bar over the symbol (indicating a multiple of 1,000..ie. C bar = 100,000) No symbol for zero, no place value. 

What are the characteristics of the IndioArabic System?

Developed in India by Hindu's, and brough to the west by the Arabs
Base 10 with symbols for the numbers 09 Uses place value 

How do you change a number from base ten to a different base?

Find out what the greatest power of the different base is that you can subtract from the base 10 number. subtract it the most amount of times you can. That goes into that position. subtract the next lower power of the new base, etc. until you just have units left.


How do you change a number from a different base to a base 10?

Figure out what the positions are (ie, units, power 1, power 2, etc) and multiply the number in that position by the base to the power of the position. Add the answer together, and it's in base 10.
