MIT Math Syllabus 10-3 Lesson 1: Sets and the real number system
Pptmath
1. Standards: A.A.29 and A.A.30
POD: Ryan Gosling tells Michael that -4 is an integer and a
whole number. Is he right or wrong? Explain why, and if not,
state all the number groups -4 belongs to.
???????
????
2. Solution
Ryan Gosling is WRONG, because -4 is an
integer, but it is not whole. Whole
numbers are {0,1,2,3,4…}, and do not
include negative numbers. -4 belongs to
rational and integer numbers.
5. Obviously, there are more Canadians
than Ontarians in the world, and not
all Canadians are from Ontario
6. But, every single person from
Ontario is from Canada
ALL CANADIANS
People from Ontario
7. So what does this mean? (Other
than your new knowledge that Ryan
Gosling is from Ontario)
The form {} In expressing sets is known as
roster form. A set is a collection of
elements or members. A set is denoted by {}.
For example: {7,9,12,16}
So, a set of all Canadian territories and
provinces is {Manitoba, Alberta, Ontario,
Saskatchewan, Quebec, Prince Edward’s
Island, New Brunswick, Nova Scotia,
Newfoundland and Labrador}
8. An example of sets is the real
number system. Natural numbers is a
subset of whole numbers, as whole
numbers include every number which is
included in natural numbers, excluding 0.
9. A subset consists of elements from any given
set. If A={1,2,3}, and B={1,3}, then B is a subset
of A. A subset is indicated by ⊆ In this
case, B⊆ A. If you want to indicate that B is not
a subset of A, you would indicate that with ⊄.
For example:
A={4,5} B={4}
B ⊆ A, because B only has some members, but not all the
members in A.
A subset of the Canadian territories can be
{Ontario}, only, or {Ontario, Quebec}. “Ryan Gosling is from
Canada, meaning {All Canadian territories}, but more
specifically, {Ontario}.
However, a subset can have as many members
which are in the original set, even 0! Subsets
with no members represented by {} or Ø are
known as null sets.
10. A power set is the set of all subsets
of a set.
You can calculate the number of
subsets in a set by using the formula
2^n, where n= the amount of terms in
a set when no members of the set
are the same.
For example:
How many subsets are in {13, 25, E, W, F, 9, 4}?
There are 7 terms in the set, so you could plug in 7 for n,
making 2^7. 2^7 is equal to 128 subsets.
12. Solution
There are 8 elements in the set. So,
using the formula 2^n, you would
substitute n for 8. 2^8 is 256, so there
are 256 different subsets.
13. However, sometimes you may be asked to
list only the PROPER SUBSETS. An
improper subset is a subset that is the
original set. Therefore, the number of
proper subsets in {13, 25, E, W, F, 9, 4}
is 127 subsets, excluding the improper
subset of {13, 25, E, W, F, 9, 4}.
14. If a set has infinite members, it is
called an infinite set. If it has a
certain amount of members, it is a
finite set.
Examples of infinite sets include
{2,4,6,8,10,12…}. Infinite sets are noted
by the “…”, known as ellipses. However,
ellipses can be used for finite sets when
you don’t feel like writing the complete
set, such as {a,b,c,d…w,x,y,z}.
16. Done yet? No? It’s infinite you say?
This is a universal set, or universe. It is the
largest set possible with the subject you
are using. It doesn’t necessarily have to
be infinite, like numbers though.
For example, a universal set could be {months of the
year} which would stand for
{January, February, March, April, May, June, July, Augu
st, September, October, November, December}. Months
aren’t infinite but 12 months is the most you can
have, so it’s universal.
17. Please note that it’s
spelled complement,
not compliment. If
we wanted a
compliment, just
comment on our
great looks
If you had a subset of the universal set,
being P = {January, May, July, August,
October, December}, them the
complement of the set would be P’ (P
prime) = {February, March, April, June,
September, November}. The complement
of the set contained elements of the
universal sets of {months of the year} not
already given in a subset.
A complement of a set is everything that
is in the universal set but not inside the
subset. In this example it’s P.
18. Try it!
Ryan Gosling has to choose what color to dye his hair. His preferred
choice is any color of the rainbow. When Ryan goes to the salon, his hair
stylist, Austin, tells him that they don’t carry indigo and violet. What colors
can Ryan Gosling dye his hair? Create a universal set, a subset of the
universal set, and find the complement of the set. Don’t forget to use
variables!
How would you like
it done sir?
19. Solution
Answer:
Let C = {colors of the rainbow} and let N
={indigo, purple}
Since Ryan cannot dye his hair indigo or
purple, the answer is the complement of N.
You would express the complement of N
using N’.
N’ = {red, yellow, blue, green, orange}
Ryan Gosling can dye his hair red, yellow,
blue, green, or orange.
20. How else could you express sets?
You could also express a set with setbuilder notation. Set builder notation
is a way of “building a set” by listing the
requirements and properties of the
elements of the set.
Example:
21. Another way to express sets is through
interval notation. An interval is a
connected subset of numbers. Interval
notation is an alternative to expressing
your answer as an inequality.
In interval notation, “(“, known as a soft
bracket indicates "not included" or
"open". “[“, known as a hard bracket
includes means "included" or "closed".
Example: (a, b] is interpreted as a < x ≤
b where the endpoints are NOT included.
22. Try it again!
Ryan Gosling is in the running for the
sexiest man alive award. His competition,
however, is Joseph Gordon-Levitt and
Benedict Cumberbatch. A panel of 200
people will vote on the winner. At the
moment, Joseph Gordon-Levitt has 64
votes, Benedict Cumberbatch has 63 votes
while Ryan Gosling has 66 votes. In set
builder notation, what number of votes
from the remaining votes does Ryan
Gosling need to become the sexiest man
alive?
23. Solution
Answer: P = {x|x is a whole number and x ≥ 3}
Out of 200 panelists, 64 of them voted for
Joseph Gordon-Levitt, 63 voted for Benedict
Cumberbatch, and 66 voted for Ryan Gosling.
So, in total, 193 voted. If all 7 remaining
panelists all voted for Joseph Gordon-Levitt or
they all voted for Benedict Cumberbatch, then
Ryan Gosling would lose. If Ryan Gosling gets at
least 3 votes, even if the other 4 voted for
Joseph Gordon-Levitt, Ryan Gosling would still
win by 1 vote.
I hope they
are not that
upset that
they lost
24. Group Work
Ryan Gosling is constructing a swimming pool in his backyard. He has already
decided that it will be 5 feet deep. He wants the length of the pool to be at least
2016 inches but no more than 3192 inches, and the width to be shorter than 150 yards
but longer than 75 yards. His agent tells him that it will cost $10 per cubic foot of
the volume of the pool he wants made, and only whole numbered dimensions will be
accepted. There will be a 15% tip required on top of the final price.
1.
2.
3.
4.
5.
6.
7.
In interval notation, describe what lengths of the pool Ryan Gosling would accept.
Make a number line to show the data. Let L = the length in feet.
In interval notation, describe what widths of the pool Ryan Gosling would accept.
Make a number line to show the data. Let W= the width in feet.
What is the greatest number of cubic feet that could be in the pool?
What is the least number of cubic feet that could be in the pool?
What is the greatest amount (tip included) that Ryan Gosling could pay for creating
the pool?
What is the least amount (tip included) that Ryan Gosling could pay for creating the
pool?
If Ryan Gosling decides he wants a square shaped pool, write in set builder notation
what possible width he could make the pool. Let x = all possible dimensions for
length and width.
`
I’d say this new pool will
set up nicely
25. Group Work Solutions
1.
L = [168, 266]
2. W = (225, 450)
597170 cubic feet
189840 cubic feet
$6,867,455.00
$2,183,160.00
{x | 226 ≤ x ≤ 266}
