10. RATIONAL NUMBERS (Q
(Q
)
numbers that can be expressed as a
quotient a/b, where a and b are integers.
terminating or repeating decimals
11. c
Any number that
cannot be written as a 3 1.732050808
ratio (fraction) 3
2 1.492106248
Any number whose e1 2.718281828
decimal 3.141592654
representation neither
repeats or stops.
12. Set of all rational and irrational
numbers.
c
A Real Number is any number that can be
graphed in the number line
13. . Graph the numbers 0.2, 7 , 1, 2 , and 4 on a
number line. 10
7
4 1 0.2 2
10
4 3 2 1 0 1 2 3 4
21. Given a real number a,
Addition:
Multiplication:
22. Given a real number a,
Addition:
Multiplication: -1
a 0
23. For any real numbers a, b, and c
Addition Multiplication
Commutative
a b b a a b b a
Associative
(a b) c a (b c) (a *b)* c a *(b * c)
Identity a+0=a=0+a a a a
If a in not zero, then
Inverse
a ( a) 0 ( a) a 1 1
a *a 1 a *a
24. Pr o p e r t i e s o f Eq u i v a l e n c e o r
Eq u a l i t y
Addition and For any reals a, b, and c, if
Subtraction a=b then a+c=b+c and
a-c=b-c
Multiplication For any reals a, b, and c, if
and Division a=b then a*c=b*c, and, if c
is not zero, a/c=b/c
25. Pr o p e r t i e s o f Or d e r o r
Inequalit y
Result:
Between any two real numbers there is a
rational number and an irrational number.
1- For any real numbers a and b
either a < b , b < a or a=b
2- If a < b and b < c then a < c
26. 3- If a < b then a+c < b+c c R
4- If a < b and c > 0 ,then ac < bc
and a b
c c
If a < b and c < 0, then ac > bc
and a b
c c
27. 5- If a < b and a = c , then c < b
6- If 0 < a < b or a < b < 0, then
1 1
a b
28. 7- If 0 < a < b and n > 0 , then
an < bn and n a < n b
29. - is an element of
~ - not or negation of
- Union
- Intersection
- is a subset of
- and (conjunctive/ intersection of two sets)
- or (disjunctive/ union of two sets)
> - Greater than & Greater than or equal to
< - Less than & Less than or equal to
= - equal and not equal to
30. Tell w hic h of t he propert ies of real
num bers just ifies eac h of t he follow ing
st at em ent s.
1. (2 )(3 ) + (2 )(5 ) = 2 (3 + 5 )
2. (1 0 + 5 ) + 3 = 1 0 + (5 + 3 )
3. (2 )(1 0 ) + (3 )(1 0 ) = (2 + 3 )(1 0 )
4. (1 0 )(4 )(1 0 ) = (4 )(1 0 )(1 0 )
5. 1 0 + (4 + 1 0 ) = 1 0 + (1 0 + 4 )
6. 1 0 [ (4 )(1 0 )] = [ (4 )(1 0 )] 1 0
7. [ (4 )(1 0 )] 1 0 = 4 [ (1 0 )(1 0 )]
8. 3 + 0 .3 3 i s a r e a l n u m b e r
60. Simplify the expressions. Express all answers
in terms of positive exponents. Rationalize the
denominator where necessary to avoid fractional
exponents of denominator.
Answer:
61. Simplify the expressions. Express all answers
in terms of positive exponents. Rationalize the
denominator where necessary to avoid fractional
exponents of denominator.
Answer:
62. Algebraic expressions are numbers
represented by symbols which are
combined by any or all of the arithmetic
operations such as addition, subtraction,
multiplication and division as well as
exponentiation and extraction of roots.
63. Algebraic expressions
with exactly one term : monomials
with exactly two terms: binomials
with exactly three terms: trinomials
with more than one term: multinomials
67. Special Products
Refer to page 18 of textbook for list of
rules for special products
Prob.19 (Sec. 0.4) Perform the indicated
Prob.19
operations and simplify
(x + 4)(x + 5)
Answer: (x +4)(x + 5) = x2 + 5x + 4x + 20
= x2 + 9x + 20
71. Factoring is rewriting expression as a
product of 2 or more factors
E.g. If c = ab, then a and b are factors of c
Refer to page 21 of textbook for list of
rules for factoring
72. Common Factors
Prob.5
Prob.5 (Sec. 0.5) Factor the following
expressions completely
8a3bc - 12ab3cd + 4b4c2d2
12ab
Answer:
4bc(2a3 - 3ab2d + b3cd2)
bc(2
74. Algebraic expressions which are
fractions can be simplified multiplying
and dividing both numerator and
denominator of a fraction by the same
non-
non-zero quantity
83. Mathematical Systems,groups and fields
a set of elements
One or more operations defined on this set
Definitions and rules for applying the operations
on the set.
Theorems can be deduced from the given
definitions and rules.
84. The set G is closed under the operation *
The operation * is associative
There is an identity element e of G for *
There is an inverse element for every
element of G
85. A Field is a mathematical system that
consisting a set F and two operations that
satisfy 11 properties:
86. Domain a set of all possible
replacement values for a given variable.
Ex. D= { x| x R}
Quantifier a word or phrase that
describes in general terms the part of the
domain for which a sentence/ statement is
true. Ex. - ,
87. Universal Quantifier a statement that has
the same truth value for every element of the
domain. X = > For all/ every x
Existential Quantifier a phrase that
describes a statement as being true for some
or at least one element from the domain.:
X : x + 3 = 7 = > There exist a value for
x such that x + 3 = 7
88. Negating Quantified Statements
The negation of a universally quantified
statement p is an existentially quantified
statement of the negation of p (~ p).
The negation of x p is expressed as x ~p
The negation of an existentially quantified
statement p is a universally quantified
statement of the negation of p (~ p).
The negation of x p is expressed as x ~p
89. X :x= 2
X :x> 7
X ~ k
Some animals can fly
Some rectangles are squares
90. Set - a collection of objects or elements.
2 types - Finite & Infinite
eg. { 1 , 2 , 3 } { 2 , 4 , 6 , 8 } { all w omen < 21 }
21}
{ }
Subset - a set whose entire contents also
belongs to another set
91. Define each of the following and show
how they are represented?
Empty Set
Union of sets
I ntersection of sets
Universal set -
Complement of a set
92. Empty Set a set with no members - { } or
Union of sets A set containing the members of both/all given given sets
Union of sets A set containing all all the members of both/all sets
Intersection of sets a set containing only members that are elements of BOTH/ALL
sets
Universal set - The complete set or groups of elements from which solution
variables/subsets can be chosen. Normally the Universal set is also the Domain.
Complement of a set If a subset A of elements is identified within Universal set U,
elements is identified within Universal set U,
the complement A is all the elements that are NOT in the identified set, but are if the
universal set.
ex. if the universal set is the set of natural numbers, and the set of even numbers is
identified, then the complement of that set is the set of odd numbers.
