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. ???????

????

SWBAT analyze, create and identify sets, subsets and ways to express them.

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.

Ryan Gosling is Canadian

Ryan Gosling is also from Ontario

Obviously, there are more Canadians than Ontarians in the world, and not all Canadians are from Ontario

But, every single person from Ontario is from Canada

ALL CANADIANS

People from Ontario

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}

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.

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.

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.

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.

Try it!

How many subsets are in {1, b, E, r, T, W, i, L,}?

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.

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}.

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}.

DRILL:

START COUNTING EVERY

SINGLE

NUMBER

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, August, September, October, November, December}. Months aren’t infinite but 12 months is the most you can have, so it’s universal.

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

Please note that it’s spelled complement, not compliment. If we wanted a compliment, just comment on our great looks

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?

SolutionLet 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.

How else could you express sets?

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.

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".

In a line graph, you would show that the endpoint is included by coloring the circle. You would show the point is not included by leaving the circle empty. Example:  (a, b]  is interpreted as a < x ≤ b  where the endpoints are NOT included.

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?

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

Group WorkRyan 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. 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.

2. 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.

3. What is the greatest number of cubic feet that could be in the pool?4. What is the least number of cubic feet that could be in the pool?5. What is the greatest amount (tip included) that Ryan Gosling could pay for creating

the pool? 6. What is the least amount (tip included) that Ryan Gosling could pay for creating the

pool?7. 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

Group Work Solutions 1. L = [168, 266]

2. W = (225, 450)

3. 597170 cubic feet 4. 189840 cubic feet 5. $6,867,455.00 6. $2,183,160.00 7. {x | 226 ≤ x ≤ 266}

168 266

225 450

with this new knowledge, you’ll be

set for our quiz tomorrow