University of Makati
J.P. Rizal Extension, West Rembo, Makati City

SETS, SET NOTATION and VENN DIAGRAM

Group IV
Abellar, Ana Mei L.
Daquioag, Juliet
Mendoza, Dither S.
Mitra, Erica Mae M.
Ocampo, Jaypee
Peralta, Mary Grace
Salig, Aerona Rea A.

Angelyn M. Pangilinan
Professor
SETS * is a well-defined collection of distinct objects. A well-defined set means that it is possible to determine whether an object belongs to a given set.
Venn Diagram
Venn Diagram

Kinds of Set 1. Equal Set 2. Equivalent Set 3. Finite Set 4. Infinite Set 5. Joint Set 6. Disjoint Set 7. Universal Set 8. Subsets

Venn diagram * Invented by John Venn * as a way of picturing relationships between different groups of things * actually called Euler diagram

Example:

Universe Disjoint Set

Joint Set

SET NOTATION 1. A U B 3. A’ or ~A

Answer: A U B Answer: A’ or ~A

2. A ∩ B 4. ( A U B) ´ or ~ ( A U B)

Answer: A ∩ B Answer: ( A U B) ´ or ~ ( A U B)

5. (A ∩ B) ´ or ~ (A ∩ B) Answer: (A ∩ B) ´ or ~ (A ∩ B)

REVIEW EXERCISES
( A U B ) ʹ ∩ C Answer: ( A U B ) ʹ ∩ C A B A B C C ( A U B ʹ ) U C Answer: ( A U B ʹ ) U C

A B A B C C

( A ʹ ^ B ) U C Answer: ( A ʹ ^ B ) U C

A B A B

C C Write down the elements in the following sets.
Let U = { 0,1,2,3,4,5,6,7,8,9,10} A = { 0,1,2,3,5,8} B = { 0,2,4,6} C = {1,3,5,7}
1. Bʹ 2. B ∩ C 3. A U C
Answer:
1. Bʹ = {1,3,5,7,8,9,10} 2. B ^ C = { } 3. A U C = { 0,1,2,3,5,7,8 }

WORD PROBLEMS 1. There were 50 people in a shoe store. 18 people were about to buy Nike shoes and 15 people were about to buy Reebok shoes? If 3 people were about to buy both brands of the shoes, how many people were in the store, but not buying shoes? 2. There are 400 students enrolled in an institute. 20% of students are majoring in engineering and 35% of students are accountancy. 15% of students are majoring in both fields. How many students are enrolled in engineering and accountancy field? 3. 100 customers took the survey test asking, “Among the 3 beverages, what would they like to drink after tasting these beverages. Eighty-eight customers chose the first beverage while ninety chose the second one. Eighty-seven customers chose the third beverage. From these customers, eighty like to drink all of these. Four customers don’t like any of these. If eighty-four of these customers like beverage A and B, and Eighty-two customers like both B and C, customers have like Beverage A and C? What is its probability?
The following examples should help you understand the notation, terminology, and concepts related to Venn diagrams and set notation.

Let's say that our universe contains the numbers1, 2, 3, and 4. Let A be the set containing the numbers 1 and 2; that is, A = {1, 2}. (Warning: The curly braces are the customary notation for sets. Do not use parentheses or square brackets.) Let B be the set containing the numbers 2 and 3; that is, B = {2, 3}. Then we have the following relationships, with pinkish shading marking the solution "regions" in the Venn diagrams: set notation | pronunciation | meaning | Venn diagram | answer | A U B | "A union B" | everything that is in either of the sets |

| {1, 2, 3} | A ^ B or | "A intersect B" | only the things that are in both of the sets |

| {2} | Ac or ~A | "A complement", or "not A" | everything in the universe outside of A |

| {3, 4} | A – B | "A minus B", or
"A complement B" | everything in A except for anything in its overlap with B |

| {1} | ~(A U B) | "not (A union B)" | everything outside A and B | | {4} | ~(A ^ B) or ~() | "not (A intersect B)" | everything outside of the overlap of A and B | | {1, 3, 4} |

Given the following Venn diagram, shade in A ^ C

The intersection of A and C is just the overlap between those two circles,

Given the following Venn diagram, shade in A U (B – C).

As usual when faced with parentheses, I'll work from the inside out.
I'll first find B – C. "B complement C" means I take B and then throw out its overlap with C, which gives me this:

Now I have to union this with A:

Given the following Venn diagram, shade in~ [(B UC) – A].

The union of B and C shades both circles fully:

Now I'll do the "complement A" part by cutting out the overlap with A:
Practically speaking, the "not" complement with the tilde says to reverse the shading

The tilde ("TILL-duh") is the wiggly "~" character at the beginning of ~ [(B UC) – A]; on your keyboard, the tilde is probably located at or near the left-hand end of the row of numbers. The tilde, in this context, says that I now want to find the complement of what I've shaded. There are two kinds of complement in this problem. The set-subtraction complement in the previous step throws out any overlap between two given sets. But the kind of complement we see in this step, the "not" complement, means "throw out everything you have now and take everything else in the universe
For instance, the picture to the right displays that A is a subset of B

As you can see above, a subset is a set which is entirely contained within another set. For instance, every set in a Venn diagram is a subset of that diagram's universe.
Venn diagrams can also demonstrate "disjoint" sets.
In the graphic to the right,A and B are disjoint:

That is, disjoint sets have no overlap; their intersection is empty. There is a special notation for this "empty set", by the way: "Ø". (Unless you have an odd computer set-up, the preceding character looks like an "O" with a forward slash through it. If you're on a PC, you can type this "empty set" character by holding down the "ALT" key and typing "0216" on the numeric keypad.) This "Ø" character is pronounced as "the empty set".
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An illustration of a use of these set relationships would be the manner in which some search engines process searches:
• If you type "cats AND dogs" into the search box, a search engine using this syntax (called "Boolean" logic) will return all web pages that contain both the word "cats" and the word "dogs". This corresponds to the set "C ^ D".
• If, on the other hand, you type "cats OR dogs", the search engine will return web pages that contain either the word "cats" or the word "dog" (or both, because the mathematical meaning of "or" is "inclusive"). This "or" statement corresponds to the set "C U D".
• If you type "cats NOT dogs", the search engine will return pages containing the word "cats", but only after discarding all the pages which also contain the word "dogs". This corresponds to the set "C – D
Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance:
Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.
a) If five students are in both classes, how many students are in neither class?
b) How many are in either class?
c) What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?
There are two classifications in this universe: English students and Chemistry students.

First I'll draw my universe for the forty students, with two overlapping circles labelled with the total in each:

Since five students are taking both classes, I'll put "5" in the overlap:
I've now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so I'll put "9" in the "English only" part of the "English" circle:

I've also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so I'll put "24" in the "Chemistry only" part of the "Chemistry" circle:

This tells me that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both). This leaves two students unaccounted for, so they must be the ones taking neither class.

From this populated Venn diagram, I can get the answers to the questions.
Two students are taking neither class.
There are 38 students in at least one of the classes.
There is a 24/40 = 0.6 = 60% probability that a randomly-chosen student in this group is taking Chemistry but not English.

Suppose I discovered that my cat had a taste for the adorable little geckoes that live in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet: Six gray geckoes, twelve geckoes that had dropped their tails in an effort to escape capture, and fifteen geckoes that he’d chewed on a little. Only one of the geckoes was gray, chewed on and tailless; If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of “gray”, “tailless”, and “chewed on”, how many were tailless and chewed on but not gray?

There was one gecko that was gray, tailless, and chewed on, so I'll draw my Venn diagram with three overlapping circles and put "1" in the center overlap:

Two were gray and tailless but not chewed on, so "2" goes in the rest of the overlap between "gray" and "tailless".

Two were gray and chewed on but not tailless, so "2"goes in the rest of the overlap between "gray" and "chewed-on".

Since a total of six were gray, and since 2 + 1 + 2 = 5 have already been accounted for, this tells me that there was only one left that was only gray.

This leaves me needing to know how many were tailless and chewed on but not gray, which is what the problem asks for. Since I don't know how many were only chewed on or only tailless, I cannot yet figure out the answer.

I'll let "x" stand for this unknown number of tailless, chewed-on geckoes.

I do know the total number of chewed geckoes (15) and the total number of tailless geckoes (12). This gives me: only chewed on: 15 – 2 – 1 – x = 12 – x

only tailless: 12 – 2 – 1 – x = 9 – x

There were a total of 24 geckoes for the month, so adding up all the sections of the diagram's circles gives me:
1 + 2 + 1 + 2 + x + (12 – x) + (9 – x) = 27 – x = 24
Solving, I get that x = 3
Three geckoes were tailless and chewed on but not gray.
(Advisory: No geckoes or cats were injured during the production of the above word problem".

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