Transcript Chapter 1 Review

Sets and Geometry
Working with Sets
A set is a collection of unique elements. Elements in a set do not "repeat".
Methods of Describing Sets:
Sets may be described in many ways: by roster, by set-builder notation, by interval notation, by
graphing on a number line, and/or by Venn diagrams.
By Roster: A roster is a list of the elements in a set, separated by commas and surrounded by French
curly braces.
is a roster for the set of integers from 2 to 6, inclusive.
is a roster for the set of positive integers. The three dots indicate that the
numbers continue in the same pattern indefinitely.
Rosters may be awkward to write for certain sets that contain an infinite number of entries.
By Set-builder notation: Set-builder notation is a mathematical shorthand for precisely stating all
numbers of a specific set that possess a specific property.
= real numbers;
= integer numbers;
= natural numbers.
is set-builder notation for the set of integers from 2 to 6, inclusive.
= "is an element of"
Z = the set of integers
| = the words "such that"
The statement is read, "all x that are elements of the set of integers, such
that, x is between 2 and 6 inclusive."
The statement is read, "all x that are elements of the set of integers, such
that, the x values are greater than 0, positive."
(The positive integers can also be indicated as the set Z+.)
It is also possible to use a colon ( : ), instead of the | , to represent the words "such that".
is the same as
By interval notation: An interval is a connected subset of numbers. Interval notation is an alternative
to expressing your answer as an inequality. Unless specified otherwise, we will be working with real
numbers.
When using interval notation, the symbol:
(
means "not included" or "open".
[
means "included" or "closed".
as an inequality.
in interval notation.
The chart below will show you all of the possible ways of utilizing interval notation.
Interval Notation: (description)
Open Interval: (a, b) is interpreted as a < x < b where the
endpoints are NOT included.
(While this notation resembles an ordered pair, in this context it
refers to the interval upon which you are working.)
Closed Interval: [a, b] is interpreted as a < x < b where the
endpoints are included.
Half-Open Interval: (a, b] is interpreted as a < x < b where a is
not included, but b is included.
Half-Open Interval: [a, b) is interpreted as a < x < b where a is
included, but b is not included.
Non-ending Interval:
is interpreted as x > a where a is not
included and infinity is always expressed as being "open" (not
included).
Non-ending Interval:
is interpreted as x < b where b is
included and again, infinity is always expressed as being "open"
(not included).
(diagram)
(1, 5)
[1, 5]
(1, 5]
[1, 5)
Union and Intersection of Sets
Union of Sets
The union of two sets, denoted AUB ("A union B"), is the set of all members contained in either A or B
or both. We can think of the union of two sets as the entire Venn diagram:
Union of Sets
Union of A and B
Example : What is XUY if X = { -4, 3, 2, 11, -6} and Y = {3, 6, 11, -4, 5}?
The easiest way to write the union of two sets is to write all the members in the first set, and then write
all the members in the second set that haven't been written yet:
XUY = { -4, 3, 2, 11, -6, 6, 5}
Example : What is AUB if A = {2, 3, 4} and B = {5, 6, 7}?
AUB = {2, 3, 4, 5, 6, 7}
The number of members in AUB should be the total number of members in A plus the total number
of members in B minus the number of members which are in both sets
Intersection of Sets
The intersection of two sets, denoted A∩B ("A intersect B") is the set of all members contained in both
A and B. We can think of the intersection of two sets as the overlap in the Venn diagram:
Intersection of Sets
Example: What is X∩Y if X = { -4, 3, 2, 11, -6} and Y = {3, 6, 11, -4, 5}?
X∩Y = { -4, 3, 11}
Example: What is A∩B if A = {2, 3, 4} and B = {5, 6, 7}?
Since A and B have no members in common, A∩B = Ø (the null set).
GEOMETRY
Formula given on the Regents:
Area of a trapezoid:
A = ½h(b1 + b2)
Surface Area of a rectangular prism: SA = 2lw + 2hw + 2lh
Volume of a cylinder:
Surface Area of a cylinder:
V = πr2h
SA = 2πr2 + 2πrh
Formula not given on the Regents
Area of a circle = πr2
Circumference of a circle = 2πr
Relative Error
Any measurement made with a measuring device is approximate.
If you measure the same object two different times, the two measurements
may not be exactly the same. The difference between two measurements is
called a variation in the measurements.
Another word for this variation - or uncertainty in measurement - is "error."
This "error" is not the same as a "mistake." It does not mean that you got the
wrong answer. The error in measurement is a mathematical way to show the
uncertainty in the measurement. It is the difference between the result of the
measurement and the true value of what you were measuring.
When the accepted or true measurement is known, the relative error is found
using:
Note that the relative error is always a positive answer – that is why the
formula uses the absolute value on the top line. If your calculation gives you a
negative answer, just write it as a positive value in the final step.
Wendy measures the floor in her rectangular bedroom for new
carpeting. Her measurements are 24 feet by 14 feet. The
actual measurements are 24.2 feet by 14.1 feet. Determine the
relative error in calculating the area of her bedroom. Express
your answer as a decimal to the nearest thousandth
Solution:
Measured value = 24 x 14 = 336 ft2
Actual value = 24.2 x 14.1 = 341.22 ft2
= 336 – 341.22
= -0.015298
341.22
Finally change the negative to a positive value and round to the
‘nearest thousandth ‘
The answer is 0.015. (Note there are no units)
1. Which interval notation represents -3 ≤ x ≤ 3?
(1) [-3,3]
(2) (-3,3]
(3) [-3,3)
(4) (-3,3)
2. Which set builder notation describes {-2, -1, 0, 1, 2, 3}?
(1) {x│-3 ≤ x ≤ 3, where x is an integer}
(2) (2) {x│-3 < x ≤ 4, where x is an integer}
(3) {x│-2 < x < 3, where x is an integer}
(4) {x│-2 ≤ x < 4, where x is an integer}
3. Given: R = {1,2,3,4}
What is R∩P?
A = {0,2,4,6}
(1) {0,1,2,3,4,5,6,7}
(2) {1,2,3,4,5,7}
P = {1,3,5,7}
(3) {1,3}
(4) {2,4}
4. Given:A = {2,4,5,7,8} and B = {3,5,8,9}
What is A U B?
(1) {5}
(2) {5,8}
(3) {2,3,4,7,9}
(4) {2,3,4,5,7,8,9}
5. A garden is in the shape of an isosceles trapezoid and a semicircle,
as shown in the diagram below. A fence will be put around the
perimeter of the entire garden.
Which expression represents the length of fencing, in meters, that
will be needed?
1) 22 + 6π
2) 22 + 12π
3) 15 + 6π
4) 15 + 12π
6. In the diagram below of rectangle AFEB and a semicircle
with diameter CD, AB = 5 inches, AB = BC = DE = FE, and CD =
6 inches. Find the area of the shaded region, to the nearest
hundredth of a square inch.
7. Oatmeal is packaged in a cylindrical container, as shown in the
diagram below.
The diameter of the container is 13 centimeters and its height is 24
centimeters. Determine, in terms of π, the volume of the cylinder, in
cubic centimeters
8. If the volume of a cube is 8 cubic centimeters, what is its
surface area, in square centimeters?
1) 32
2) 24
3) 12
4) 4
9. Students calculated the area of a playing field to be 8,100
square feet. The actual area of the field is 7,678.5 square feet.
Find the relative error in the area, to the nearest thousand
10 To calculate the volume of a small wooden cube, Ezra measured
an edge of the cube as 2 cm. The actual length of the edge of
Ezra’s cube is 2.1 cm. What is the relative error in his volume
calculation to the nearest hundredth?
1) 0.13
2) 0.14
3) 0.15
4) 0.16