This document discusses sets, subsets, and different ways to represent them. It defines a set as a collection of elements, and a subset as a set that contains elements from another set but not necessarily all elements. Sets can be represented through roster form using curly brackets, set-builder notation describing the properties of elements, and interval notation for numbers. The document provides examples of sets, subsets, universal sets, complements, and using set notation to solve problems about voting outcomes.

1. SWBAT analyze, create and
identify sets, subsets and ways
to express them.
POD:???
Standards: A.A.29 and A.A.30

2. Congratulations!!!! You have just identified
examples of sets and subsets.

3. Ryan Gosling is Canadian

4. Ryan Gosling is also from Ontario

5. Obviously, there are more Canadians
than Ontarions 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)
A set is a collection of elements or
members. A set is denoted by {}.
For example: {3,4} {2} {7,9,12,16}
So, a set of all Canadian territories and
provinces is
{Manitoba, Alberta, Ontario, Saskatchewa
n, Quebec, Prince Edward’s Island, New
Brunswick, Nova Scotia, Newfoundland
and Labrador}

8. 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, A⊂B.
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
{Manitoba, Alberta, Ontario, Saskatchewan, Quebec, Prince Edward’s
Island, New Brunswick, Nova Scotia, Newfoundland and Labrador}, but more
specifically, {Ontario}.
However, a subset can have as many members
which are in the original set, even 0! Subsets
with no members is represented by {} or Ø
are known as null sets.

9. 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
extremely simple formula 2^n, where
n= the amount of terms in a set.
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.

10. Try it!
How many subsets are in {1, b, E, r, T, W, i, L,}?

11. However, sometimes you may be asked to
list only the PROPER SUBSETS. What
are proper subsets? Proper subsets are
subsets that do NOT include all
elements in a 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}.

12. 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}.

13. DRILL:
START COUNTING
EVERY
SINGLE
NUMBER

14. Done yet? No? It’s infinite you say?
This is a universal set, or universe 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.

15. If you had a subset of the universal
set, being P =
{January, May, July, August, October, Dec
ember}, 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.

16. LETS TAKE A LOOK AT A REAL
WORLD EXAMPLE!
In this
diagram, the
universal set is
Canadians while the
subset is Ontarians.
Complement of
Ontarians is
Canadians that are
not Ontarians
which is
{Alberta, British
Columbia, Saskatch
ewan, Manitoba…Qu
ebec}
ALL CANADIANS
People from Ontario

17. 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 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?

18. 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.

19. How else could you express sets?
The form {} In expressing sets is known
as roster form. You could also express
a set with set-builder notation.Set
builder notation is a way of “building a
set” by listing the requirements and
properties of the elements of the set.
Example:

20. 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.
[a, b] is interpreted
as a < x < b where the endpoints are
included.

21. Try it again!
Ryan Gosling is in the running for the sexiest
man alive award. His
competition, however, is Joseph GordonLevitt 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?

22. 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.

23. GROUP WORK
?