Math 116 Exam 1
e
"element of" ex. 6eZ
|
"such that"
whole numbers
0 and all positive values ex. {0,1,2,3,4,5...}
Binary numbers are base 2 numbers are made up __________
0's and 1's
facts about primes
1) 2 is the only even prime number 2) infinite number of primes
English logician john Venn developed Venn diagrams that are used to illustrate sets and relationships between sets.
1) In a Venn diagram, the universal set is represented as a rectangular region 2) subsets of the universal set are represented by oval or circular region within the universal set
2 ways to designate a set
1) describe the set using words 2) list the elements of the set inside a pair of {} ( called roaster method)
Rules employed in the roman numeration system
1) moving left to right, if the numerals are listed largest to smallest, just add the numerals 2) the numerals that are powers of 10 (I, X, C, and M) may be repeated up to three times. 3) the numerals I, X, and C are subtracted from the number to the right if their value is less than the number to the right. ex. IV= 5-1+4
basic number sets
1) real # 2) irrational # 3) rationals # 4) integers 5) whole # 6) natural/ counting #
Ex. of designating a setS,M,T,W,TR,F,SA
1) using words: the set of all the days in a week 2) roster method: {S,M,T,W,TR,F,SA}
Find the base 10 decimal representation of the binary number 100(base 2)
1*2^2+ 0*2^1+ 0*2^0 4+0+0=4
find the base 10 decimal reppresentaion of the binary number 1011000(base 2)
1*2^6+0*2^5+1*2^4+1*2^3+0*2^2+0*2^2+0*2^1+0*2^0 64+0+16+8+0+0+0= 88
In the roman numeration system, a bar over a numeral is used to denote a _______ times the value of the numeral
1000
CM
1000-100= 900
convert 687 to base five
100222(base 5) refer to notes
write CLXVIII as a hindu-arabic numeral
168
convert the numeral 201 (base three) to base 10
2*3^2+0*3^1+1*3^0 18+0+1=19
Write MMDXCIV
2594
n(A)=15 n(B)=32 n(AUB)=41 find n(A(intersect)B)
41=15+32-x x=6 n(A(intersect)B)=6
p(A)=44% p(B)=15% p(A(intersect)B)=4% find p(AUB)
44%+15%-4%=55%
CD
500-100= 400
write VII (line over top) CCLIV as a Hindu-Arabic numeral
7,254
determine the prime factorization of 380
=2*2*5*19
determine the prime factorization of 3267 in canonical
=3^3*11^2
convert 77 to binary
=64+13 =64+8+5 =64+8+4+1 2^6, 2^3, 2^2, 2^0 1001101 (base 2)
number of subsets of a set
A set with "n" elements, has 2^n subsets a. {1,2,3,4,5,6} 2^6= 64 b.{4,5,6,7,8....15} 2^12=4096 c. {} 2^0=1 subset {}
U= {0,1,2,3,4,5,6,7,8,9} A= {1,3,4,5,7} B= {2,3,4,5,6} C= {0,2,4,6,8,9} A' B' C' (A(intersect)B)' A'(intersect)B' A'UB' AU(B(intersect)C)
A'= {0,2,6,8,9} B'= {0,1,7,8,9} C'= {1,3,5,7} (A(intersect)B)'= {0,1,6,2,7,8,9} A'(intersect)B'= {0,8,9} A'UB'= {0,1,2,6,7,8,9} AU(B(intersect)C)= {1,2,3,4,5,6,7}
write the hindu-Arabic numeral 352 as a roman numeral
CCCLII
convert the numeral EA3 (base 16) to base 10
E*16^2+A*16^1+3*16^0 14*16^2+10*16^1+3*16^0 3584+160+3= 3747
Is A a proper subset of B? A- {a,e,i,o,u} B- {e,i,o,u,a}
No A=B
example of a complement set u= [1,2,3,4,5,6,7,8,9,10) S= {2.4.6.7}
S'= {1,3,5,8,9,10}
proper subset
Set A is a proper subset of set B, denoted AcB, if every element of A is an element of B and A cannot equal B
a subset of a set
Set A is a subset of set B, denoted Ac (with a line under it) B, iff every element of A is also an element of B
the complement of a set
The complement of set A, denoted by A'. is the set of all elements of the universal set, u, that are not elements of A.
write 8071 as a roman numeral
VIII (line over top) LXXI
Is N a subset of I N-{1,2,3,4,5...} I-{-2,-1,0,1,2,3,4...}
Yes NcI N and I are NOT equal
prime number
a natural number greater than one that has exactly two factors-- itself and one the only even prime number is 2
composite number
a natural number greater than one that is not prime
finite set
a set in which a number of elements are a whole number. set amount
set builder notation
another method of representing a set. especially useful when describing infinite set ex. the set of natural numbers greater or equal to four {x|xeN and X> (and equal too) 4}
Set
any group or collection of objects
Prime factorization of number using exponential notation with the factors arranged in order of increase magnitude is called the _______ factorization of the number
canonical
the complement of the _______ set is the empty set and the complement of the ______ set is the universal set.
complement, empty u'={} {}=u'
the fundamental theorem of arithmetic
every natural number larger than one is a product of primes
The inclusion-exclusion principle
for all finite set A and B, n(AUB)= n(A)+n(B)-n(A(intersect)B) n(A) represents the number of elements in the set A
The percent inclusion-exclusion principle
for all finite sets A and B p(AUB)= p(A)+p(B)-p(A(intersect)B)
The egyptian numeration system uses pictorial symbols called ________ as to represent numbers
hieroglyphics
determine if 839 is prime
it is prime. stop after 28
square root test for determining prime numbers
let n>1, neN, if no prime number less than the squre root is a divisior of n, then n's prime
numbers with bases larger than 10 need additional digits that are added starting with the _______
letter A and adding subsequent letters when needed
determine if 1073 is prime
no it is not a prime number. it id divisible by 29. it is composite
is 10,000,024 prime?
no, it is an even number
...
numbers continue in that set pattern
elements or members
objects that belong in the set
The number of digits a number of base a can have is always _________ than a
one less
integers
positive and negative whole numbers ex. {...-3,-2,-1,1,1,2,3...}
empty set is a subset of any____
set
equal sets
set A is EQUAL to set B, denoted A=B iff (if and only if) A and B have exactly the same elements ex. {a,b,c) = (c,b,a) yes
equivalent set
set A is equivalent to set B, denoted A-B, iff, A and B have the same number of elements ex. {r,u}- {v,t} yes but not equal
well defined sets
set where it is possible to determine whether any given item is an element of the set ex. a set of letter in the English alphabet
natural or counting numbers
start at 1 in a positive direction ex. {1,2,3,4,5...}
De morgan's laws: for all sets A and B
the complement of the union of two sets is the intersection of the complement of the sets. the complement of the intersection of the two sets is the union of the complement of the sets (AUB)'=A' (intersect) B' and (A(intersect) B)'= A'UB'
intersection of sets
the intersection of sets A and B, denoted A(upside down U)B, is the set of elements common in both A AND B A(upside down U)B= {x|xeA and xeB} A={1,2,3} B= {3,6,1} A(upside down U)B= {1,3}
divisor or factor
the natural number "a" is a divisor or factor of the natural number "b", provided there exist a natural number j such that aj=b ex. 12 divisors: 1,2,3,4,6,12 factors are what can be mutiplied to get 12
cardinal number
the number of elements in a finite set. denoted: n(A) ex. A= [a,e,i,o,u} n(A)=5
describe set verbally: A (upside down U) (BUC')
the set of all elements that are in A and are in set B or C complement
describe set verbally: L'UM
the set of all elements that are in L complement or are in Set M
irrational numbers
the set of all non-terminating, non-repeating decimals. ex. pie, e,
real numbers
the set of all rational and irrational numbers. Largest set
rational numbers
the set of all terminating, or repeating decimals. ex. 1/2=.5, 1/4=.25, .272727
empty set or null state
the set that contains no elements {} ex. the set of natural numbers that are negative integers
Union of sets
the union of sets A and set B, denoted AUB, is the set that contains all the elements that belong to A OR to B OR to both AUB= {x|xeA or xeB}
Every set is a subset of _______
themselves
Disjoint
two sets are disjoint if their intersection is the empty set
is 127 prime?
yes, it is a prime number. stop checking at 11
the set of primes that are less than 100
{2,3,5,7,11,13, 17, 19, 23, 29,31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}
the set of the ten smallest composite numbers
{4,6,8,9,10,12,14,15,16,18
