ACT245H1S NOTES AND EXAMPLES
©S. Broverman, 2002CALCULUS REVIEW
SET THEORY
A is a collection of . The phrase set elements " %is an element of " (% (is denoted by , and "is not an element of " %(%¤(is denoted by .
A set may be defined in terms of certain attributes of its elements, for example, the set of all odd,positive integers may be written as .
¸%O % ! # ! ¹Subset of a set: means that each element of the set is an element of the set . ( )())may contain elements which are not in .
(Union of sets: is the set all elements in either or .
(r )()(r )~¸%O % ( % )¹
Intersection of sets: is the set of all elements in both and .
(q )()(q )~¸%O % (% )¹
and The difference of two sets: (c )~¸%O % ( %¤)¹consists of all elements that are in but not in . The consists of all elements , and is () complement of the set B not in )denoted or . .
)) ))~¸%O %¤)¹c c Z ÁNote that .
(c )~(q )Z Example 1: Verify the following set relationships:
(i) (the complement of the union of and is the intersection of the ²(r )³~(q )()Z Z Z complements of and )
()(ii) (the complement of the intersection of and is the union of the ²(q )³~(r )()Z Z Z complements of and )
()Solution: (i) Since the union of and consists of all points in either , any point not in ()()or (r )()( is in neither nor , and therefore must be in both the complement of the and complement of - the intersection of and .. The reverse implication holds in a similar way )()c Z Z - if a point is in the intersection of and then it is not in it is not in so it is not in ()()Ác Z Z and (r )²(r )³²(r )³(q ), and therefore it is in . Therefore, and consist of the same Z Z Z Z collection of points - they are the sam
e set.
(ii) The solution is very similar to (i). U
Empty set:empty set The is the set that contains no elements, and is denoted . It is also J referred to as the . Sets and are called if .
null set disjoint sets ()(q )~J Some special sets of numbers are
, the ,
h ~¸ Á Á ÁÀÀÀ¹positive integers , the ,
t ~¸ Áf Áf ÁÀÀÀ¹integers ,are integers and , the k ~¸O £ ¹ rational numbers
, the ( , and are real numbers that are not rational)l real numbers l complex numbers ]~¸ b O Á ¹ ~c real numbers is the set of , where l
Example 2: For each positive rational number , let denote the set of all positive integers
5² ³ 5²³ such that is rational. How many elements are in the set ?
° A. 1 B. 2 C. 3 D. 4 E. Infinitely many
Solution : is a positive integer and is rational .
5²³~¸ O ²³¹ ° ²³~¬ 5²³²³~¬ 5²³ ° ° 1 , 2 ,4²³¬ ¤5²³²³~¬ 5²³ ° ° is not rational , ,
²³ ~ ° ° ° is irrational for because is irrational for .
Answer: C U
Relationships involving sets:
1. (r )~)r (Â(q )~)q (Â(r (~(Â(q (~(
2. (r ~(Â(q ~Â(c ~(
3. (q ²)r *³~²(q )³r ²(q *³
4. (r ²)q *³~²(r )³q ²(r *³
5. If , then and ( )(r )~)(q )~(
6. For any sets and , and ()(q ) ( (r )(q ) ) (r )
7. and ²(r )³~(q )²(q )³~(r )Z Z Z Z Z Z
8. For any set , ( (
9. h t k l ]
INTERVALS, INEQUALITIES, ABSOLUTE VALUE
AND UPPER AND LOWER BOUNDS
Open interval: Closed interval:² Á ³~¸%O % ¹´ Á µ~¸%O % ¹
Half-open interval: ´ Á ³~¸%O % ¹² Á µ~¸%O % ¹ or
Infinite interval: ´ ÁB³~¸%O %¹²c BÁ ³~¸%O % ¹
or Rules and relationships for inequalities:
1. If and , then (also, and )
¬ 2. If , then for any number , (also for )
b b 3. If and , then (also for )
4. If and , then (also for )
5. If 0, then £
6. If then has the same sign as £
7. If and have the same sign and , then
8. If , then
9. If , then
10. If , then if (and only if) & c & l l 11. If , then if (and only if) either or & & & c
l l Example 3: What is the solution set for the inequality ?
% c% Solution : .
% ¯c %~ c %b% c% c% c% c%²%c ³
~The numerator of this fraction is always , except when , so the inequality is satisfied if %~ the denominator,, is and . c % %£ U
Absolute value of a real number : %O%O ~%% O%O ~c %%
if , and if Rules and relationships for absolute values (all numbers are real numbers):
1. O O ~O c O
2. c O O O O
3. and l ~O O O O ~
4. and if then O O ~O O h O O £ ~e e O O O O
5. If , then if (and only if) , and O&O c & & ¯O&O l
6. If then if (and only if) either or ÁO&O & & c
(i.e., ) , and & ²c BÁc ³r ² ÁB³& ¯O&O l 7. and O b O O O b O O O c O O O c O O
Upper bound of a set of real numbers:
) is an of if for every ,
")% "% )upper bound Lower bound of a set of real numbers:
) is a of if for every M )M %% )
lower bound Least upper bound of a set of real numbers:
) is the of if (i) is an upper bound of , and
")")least upper bound (ii) if is any other upper bound of ;
" $$) it is denoted or ;
À"À À²)³ " ¸%O % )¹ any number 10 is an upper bound of the interval but 10 is the l.u.b. ² Á ³Greatest lower bound of a set of real numbers:
) is the of if (i) is a lower bound of , and
M )M )greatest lower bound (ii) if is any other lower bound of ;
M ) it is denoted or ;
À À À²)³ ¸%O % )¹ Example 4: Let be a set of real numbers such that and and let
: À"À À²:³~ À À À²:³~ ;~¸c O % :¹ À"À À²;³ % . Find the .
Solution : consists of numbers between and , so that consists of numbers between
:; c c c ~c À"À À²;³~c ~h h and . Then . U
Example 5: Which of the following are equal to the decimal number 15?
I. 1111 (base 2) II. 120 (base 3) III. 30 (base 5)
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
E. The correct answer is not given by A, B, C or D
Solution : I. b b b ~ b b b ~
II. b d b d ~ b b ~
III. Answer: D d b d ~ b ~ U
ANALYTIC GEOMETRY IN AND l l
Distance between two points and :²%Á&³²%Á&³ ~²%c %³b ²&c &³
l Slope of line joining points and : ²%Á&³²%Á&³ ~%£%&c&
%c% if Intercepts: the -intercept of a line is the -coordinate of the point of intersection of the line %%with the -axis and a similar definition is made for the -intercept
%&Equations of a line:
1. Point-Slope: given on a line with slope , ²%Á&³ &c &~ ²%c %³
2. Point-Point: given and on a line, ²%Á&³²%Á&³ &c&&c&%c%%c%
~ 3. Slope-Intercept: given slope and -intercept , & &~ %b
4. Intercept-Intercept: given and intercepts and , %& b ~
% &
5. General: any straight line can be put in the form (%b )&b *~ Example : Find the equation of the line between the points and and express it ²c Á ³² Á ³in the 5 ways described above. Find the - and -intercepts.
%&Solution : The equation of the line in point-point form (form 2.) is &c %c²c ³ c²c ³ c ~~c . This can be written in each of the other forms:
1 - Point-Slope: &c ~c ²%b ³
3 - Slope-Intercept: &~c %b
4 - Intercept-Intercept: % ° &b ~
5 - General: %b &c ~
The - and -intercepts are (found from the equation of the line by substituting in %&%~ &~ and solving for ) and .
%&~ U Slope of perpendicular: if a line has slope , then a perpendicular line has slope .
c The perpendicular line is also referre
d to as th
e normal line at that point.
Distance from point to line : ²%Á&³(%b )&b *~ ~O(%b)&b*O (b)
l Equation of a circle with center and radius : ² Á ³ ²%c ³b ²&c ³~
Symmetry of a graph: A graph is symmetric with respect to the -axis if the point is %²%Ác &³on the graph whenever the point is on the graph; a graph is symmetric with respect to the ²%Á&³&²c %Á&³²%Á&³-axis if is on the graph whenever is on the graph;
a graph is symmetric with respect to the origin if is on the graph whenever is ²c %Ác &³²%Á&³on the graph.
soa
Example 7: What type of symmetry is found in the graphs of and ?
&~%&~% Solution : Since for any real number , it follows that the graph of is ²c %³~%%&~% symmetric with respect to the -axis. Since , it follows that the graph of &²c %³~c %&~% is symmetric with respect to the origin.
U Example 8: Find the number of solutions to the system of equations
and .
&c %&c O%O&b %O%O ~ %b &~ Solution : The first equation can be written in the form , so that
²&c O%O³²&c %³~ the solution to the first equation is or . Substituting this into the second equation results
&~%O%O in , or equivalently, . The solutions are then %b %~ %~f ²Á³Á²Á³
c c
l l l l l and , note that is not a solution. ²Á³²Á³c c l l l l U Points in l are denoted by in rectangular coordinates. The distance between the ²%Á&Á'³points and is .
²%Á&Á'³²%Á&Á'³²%c %³b ²&c &³b ²'c '³ l A plane in 3-dimensional space can be represented in the form .
(%b )&b *'b +~
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