Math midterm Flashcards
Equal set
two sets A and B are equal if the sets contain exactly the same elements A={2,4,6} B= {2,4,6} A=B
Equivalent Set
Two sets are equivalent (A<–B) if they have the same number of elements A={1,2,3} B={4,5,6} A<–>B
Empty set
Set that has no elements represented by: S={} lSl (cardinality) = 0
0 (line through it)
Empty set - set with no elements
0
a number
{0}
set containing one element that element is the number 0
{0} - line thu 0
Set containing the symbol 0 (line through)
Universal Set
Set of all possible outcomes containing every element ex: a fair die S= {1,2,3,4,5,6} ex: S=(a deck of cards} U={a deck of cards} H={hearts}
Deck of cards cardinality
lSl = 52
Universal set lUl = 52
Hearts lHl = 13
Spades lSl = 14
H (line on top) Compliment of a set
All others not in set. If, lUl = 52 and lHl = 13 then lHl(line on top) = 52-13 lHl=39
A is a subset of B written A<(underline)B if
every element of A is also an element of B
Improper subset
Subset that is equal to original set, only 1, {5,7} is and improper subset of {5,7}
Union AUB - set of elements containing elements from A and B
Remove duplicates, list smallest to greatest: Venn diagram all parts A all parts B
Proper Subset (C)
Subset that is not equal to the original set. {5},{7}, {}, {5}c{5,7]}
A n B - intersection
what a and b have in common
Union of compliments : A(line on top) U B (line on top)
Everything that compliments A plus everything that complements B
Compliment of a union AUB (line over)
Everything that compliments A+B as a while one union
Absolute value of integers
detonated by lXl, l-5l = 5 l6l = 6
Positive + Positive
Positive
Negative + negative
negative
Positive + negative
sign of larger
negative + positive
absolute value
Positive x positive
positive
positive x negative
negative
negative x positibe
negative
negative x negative
positive
0/x =
0 for all numbers
x/0
undefined (can not divide by 0)
Closure Property
When we add or multiply any two or more integers, we obtain an integer.
Is the set of natural numbers closed for addition?
yes, pick any 2 numbers in the set and add them, you get a number in the set its closed for addition
Is the set A={1,2,3,4,5,6,7,8,9,10} closed for addition?
no, if you add 10+1 you would get 11 which is out of the set
Is the set of natural numbers N={1,2,3,4…} closed for multiplication?
Yes, if you multiply any 2 numbers you will get a number back in the set
Is the set A={0,1} closed for multiplication?
yes, 0x1=0 1x1=1 0x0=0
Is the set z = {…-3,-2,-1,0,1,2,3..} closed for subtraction?
yes
Is the set of N={1,2,3,4..} closed for subtraction?
no
The set of integers Z={..-1,2,3,0,1,2,3..} is closed for
addition, multiplication, subtraction. If you add, subtract or multiply any 2 integers you will always get an integer
Is the set of N={1,2,3,4..} closed for division?
The set of integers in not closed for division
if you divide any 2 integers you will not always get an integer
Commutative property for addition
states that the order in which 2 or more numbers are added makes no difference. property of order
Associative property for addition
allows us to group numbers for addition
Does the commutative property hold for subtraction in the set N?
no, 3-2=1, 2-3= -1
Distributive property for multiplication over addition
multiplication is distributive over addition k(m+n) = km + kn
In the set of N={1,2,3,4..} is multiplication distributive over addition?
Yes
2(2+4) = 2(7) = 14
2(3) + 2(4) = 14
In the set of natural numbers is addition distributive over multiplication?
No
3 + (4x5) = 3+ 20 = 23
(3+4) x (3+5) = 56
Use the distributive property to simplify 9 x 71
9(70 +1)
9(70) + 9(1)
630 + 9 = 639
Any integer raised to the power of 0,
is 1
a^m+n =
A^m x A^n
(ab)^n
a^n x b^n
a^1 =
a
a^0 =
1
Base 2
only has two digits (0,1)
Addition of integers order
AACAAD - Associative, associative, commutative, associative, associative, distributive
Base 10
5^10, 6^10
Base 5 - 32102
5^4 5^3 5^2 5^1 5^0
Counting in binary - 0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
11
1011
Given base 10, convert to base 2
Keep dividing by 2(or whatever base you are converting to) keep track of remainders. 8/2 = 4/2 = 2/2 = 1/2. R- 0, 0, 0, 1. Base 2 = 1000(subscript 2)
Convert 10 to binary
10/2=5 r 0
5/2 = 2 r 1
2/2 = 1 r 0
1/2 = 1 r 1
Binary = 1010(subscript 2)
0+1 =
1+0=
1
1+1=
10
1+1+1=
11
base 3 digits
0,1,2
base 4 digits
0,1,2,3
base 5 digits
0,1,2,3,4
Number of divisors property
Every natural number greater than 1 has at least 2 divisors, itself and 1
Prime numbers
Any natural number, greater than 1 that has exactly 2 divisors (1 and itself)
Composite numbers
A natural number that has more than two divisors
Is 1 a prime number?
no
What is the smallest prime number?
2
How many numbers less than 10 are prime?
2,3,5,7
Is 113 prime? *try to divide the number by all the small primes up to its square root
square of 113 between 10 and 11
only have to check prime numbers up to 10
2 no
3 no
5 no
7 no
is 114 prime? yes only has two divisors
Eratosthenes Method - finding how many numbers less that 100 are prime
Write down numbers 1-100
Cross out 1 since its not a prime number
Draw a circle around 2 and cross out every multiple of 2
Repeat step 3 for the next prime # 3
Repeat for next prime which is 5
repeat with 7
since 7 is the largest prime number less than the square root of 100=10, we know the remaining numbers are prime
Fundamental theorem of arithmetic
Every natural number greater than 1 is either a prime or a product of primes and its prime factorization is unique
Canonical Representation
Representation of an number as a product of consecutive primes using exponential notation with the factors arranged in order of increasing magnitude
Least common multiple
of a set of numbers is the smallest number that each of the numbers divides into evenly
Procedure for finding LCM
write factorizations in canonical form
select the representative of each factor with largest exponent
Multiply the representatives to find LCM
Greatest common multiple
is the largest number that divides evenly into each of the numbers in the give set
Finding the GCF
Write the factorizations in canonical form.
Select the representative of each factor with the smallest exponent.
Multiply the representatives to find the LCM.
GCD
greatest common divisor
Using Euclidean algorithm to find GCD
Start by dividing small number into larger
Keep dividing, keep track of remainders
Stop when remainder is 0
The last divisor is the GCF
