5th May 2014 TLM - Mathematics
CLASS XI
AN INTRODUCTION TO SET THEORY
Scottish Church College | Madhurima Chandra
Introduction to Set Theory
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Introduction to Set Theory
Contents SETS: ........................................................................................................................................................ 3 NUMERICAL SETS: ............................................................................................................................... 4 UNIVERSAL SET: ................................................................................................................................... 5 EQUALITY: ............................................................................................................................................. 5 SUBSETS: ................................................................................................................................................ 6 PROPER SUBSETS: ................................................................................................................................ 8 EMPTY SET (NULL SET): ..................................................................................................................... 9 ORDER: .................................................................................................................................................... 9 UNION: ................................................................................................................................................... 10 INTERSECTION: ................................................................................................................................... 11 DIFFERENCE: ....................................................................................................................................... 12 COMPLEMENT: .................................................................................................................................... 17 SOLVED PROBLEMS: .......................................................................................................................... 19 EXERCISE: ............................................................................................................................................ 20
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Introduction to Set Theory
SETS: Sets of objects, numbers, departments, job descriptions, etc. are things that we all deal with every day of our lives. Mathematical Set Theory just puts a structure around this concept so that sets can be used or manipulated in a logical way. The type of notation used is a reasonable and simple one. For example, the items you wear is a set: these would include shoes, socks, hat, shirt, pants, and so on. You write sets inside curly brackets like this: {socks, shoes, pants, watches, shirts, …}
A mathematical set is a collection of distinct objects, normally referred to as elements or members. Sets are usually denoted by a capital letter and the elements by small letters. How to write SETS: There is a fairly simple notation for sets. You simply list each element, separated by a comma, and then put some curly brackets known as "set brackets" around the whole thing. The three dots … are called an ellipsis, and mean "continue on".
The set {socks, shoes, watches, shirts, …} we call an infinite set.
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Introduction to Set Theory
The set {index, middle, ring, pinky} we call a finite set. But sometimes the "..." can be used in the middle to save writing long lists: Example: the set of letters: L={a, b, c, ..., x, y, z} In this case it is a finite set (there are only 26 letters)
NUMERICAL SETS: So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers? Set of even numbers: E={..., -4, -2, 0, 2, 4, ...} Set of odd numbers: O={..., -3, -1, 1, 3, ...} Set of prime numbers: P={2, 3, 5, 7, 11, 13, 17, ...} Positive multiples of 3 that are less than 10: N={3, 6, 9} And the list goes on. We can come up with all different types of sets. There can also be sets of numbers that have no common property, they are just defined that way. For example: {2, 3, 6, 828, 3839, 8827} {4, 5, 6, 10, 21} {2, 949, 48282, 42882959, 119484203} When we say an element a is in a set A, we use the symbol ∈ to show it. And if something is not in a set use ∉. A= {1, 2, 3}. You can see that 1 ∈ A, but 5 ∉ A Why are Sets Important? Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when you apply sets in different situations do they become the powerful building block of mathematics that they are.
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Introduction to Set Theory
Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: “Sets�.
UNIVERSAL SET:
Universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to your question.
The universal set, when doing Number Theory is the set of integers, as Number Theory is simply the study of integers.
However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.
EQUALITY: Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely! Suppose , two sets A and B where A is the set whose members are first four positive whole numbers B ={4,2,3,1} Let's check. They both contain 1. They both contain 2 and 3, and 4.So we have checked every element of both the sets, so: Yes, they are! And the equals sign (=) is used to show equality, so you would write:
A=B
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Introduction to Set Theory
SUBSETS: A set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. So, if we take some pieces of a set to form a subset. C⊆B⊆A So for example, we have the set S={1, 2, 3, 4, 5}. A subset of S is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general: A is a subset of B if and only if every element of A is in B. So let's use this definition in some examples. 1: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}? 1 is in A, and 1 is also in B 3 is in A and 3 is also in B. 4 is in A, and 4 is also in B. So all the elements of A are in B Yes, A is a subset of B Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A. Let's try a harder example. Example2: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A? Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them. The sets are: •
A = {..., -8, -4, 0, 4, 8, ...}
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Introduction to Set Theory
•
B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}
By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:
A is a subset of B, but B is not a subset of A
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Introduction to Set Theory
PROPER SUBSETS: Let A be a set. Is every element in A an element in A? yes of course, right! So wouldn't that mean that A is a subset of A? This doesn't seem very proper, does it? We want our subsets to be proper. So we introduce (what else but) proper subsets. A is a proper subset of B if and only if every element in A is also in B or A is contained in B, and there exists at least one element in B that is not in A.
Example: {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}. It is an improper subset of {1, 2, 3}.
{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set. You should notice that if A is a proper subset of B, then it is also a subset of B. When we say that A is a subset of B, we write A ⊆ B. Or we can say that A is not a subset of B by A ⊄ B ("A is not a subset of B") When we talk about proper subsets, we take out the line underneath and so it becomes A ⊂ B or if we want to say the opposite, A ⊄ B
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Introduction to Set Theory
EMPTY SET (NULL SET): This is probably the weirdest thing about sets. As an example, think of the set of piano keys on a guitar." But wait!" you say, "There are no piano keys on a guitar!" And right you are. It is a set with no elements. This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero. It is represented by ÎŚ Or by {} (a set with no elements) Some other examples of the empty set are the set of countries south of the south pole.
ORDER: No, not the order of the elements. In sets it does not matter what order the elements are in. Example: {1,2,3,4) is the same set as {3,1,4,2} When we say "order" in sets we mean the size of the set. Just as there are finite and infinite sets, each has finite and infinite order. For finite sets, we represent the order by a number, the number of elements. Example, {10, 20, 30, 40} has an order of 4. For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets. Now let's say that alex, casey, drew and hunter play Soccer: Soccer = { alex, casey, drew, hunter} (The Set "Soccer" is made up of the elements alex, casey, drew and hunter).
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Introduction to Set Theory
And casey, drew and jade play Tennis: Tennis = {casey, drew, jade}
You could put their names in two separate circles:
UNION: You can now list your friends that play Soccer OR Tennis. This is called a "Union" of sets and has the special symbol âˆŞ: Soccer âˆŞ Tennis = {alex, casey, drew, hunter, jade} Not everyone is in that set ... only your friends that play Soccer or Tennis (or both). We can also put it in a "Venn
Diagram":
Venn Diagram: Union of 2 Sets
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Introduction to Set Theory
A Venn Diagram is clever because it shows lots of information: ●
Do you see that alex, casey, drew and hunter are in the "Soccer" set?
●
And that casey, drew and jade are in the "Tennis" set?
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And here is the clever thing: casey and drew are in BOTH sets!
INTERSECTION: "Intersection" is when you have to be in BOTH sets. For ex: Pizza slice is common in a triangle and a circle.
In our case that means those who play both Soccer AND Tennis... which is casey and drew. The symbol used for Intersection is an upside down "∪" which is like this: ∩
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And this is way we represents it : Soccer ∩ Tennis = {casey, drew} In a Venn Diagram:
Venn Diagram: Intersection of 2 Sets
What is the direction of "∪"?
Think of them as "cups": ∪ can hold more water than ∩, Is it right? So, Union ∪ contains more elements than Intersection ∩
DIFFERENCE: You can also "subtract" one set from another. For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis... which is alex and hunter. And this is how we write it down: Soccer−Tennis = {alex, hunter}
In a Venn Diagram:
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Introduction to Set Theory
Venn Diagram: Difference of 2 Sets
Three Sets You can also use Venn Diagrams for 3 sets. Let us say the third set is "Volleyball", which drew, glen and jade play: Volleyball = {drew, glen, jade} But let's be more "mathematical" and use a Capital Letter for each set: ●
S means the set of Soccer players
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T means the set of Tennis players
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V means the set of Volleyball players
The Venn Diagram is now like this:
Union of 3 Sets: S ∪ T ∪ V
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Introduction to Set Theory
Now for example that: drew plays Soccer, Tennis and Volleyball jade plays Tennis and Volleyball alex and hunter play Soccer, but don't play Tennis or Volleyball no-one plays only Tennis We can now have some fun with Unions and Intersections ...
This is just the set S:S = {alex, casey, drew, hunter}
This is the Union of Sets T and V: T âˆŞ V = {casey, drew, jade, glen}
This is the Intersection of Sets S and V S ∊ V = {drew}
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Introduction to Set Theory
And now , ● take the previous set S ∩ V ● then subtract T:
This is the Intersection of Sets S and V minus Set T: (S ∩ V) − T = {} Hey, there is nothing ! Don’t worry that’s OK, it is the "Empty Set". The Empty Set has no elements: {} Sadly, one more interesting thing is that the symbol of Universal set is the letter "U" ... which is easy to confuse with the ∪ for Union. In our case the Universal Set is our Ten colleagues. U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade} We can show the Universal Set in a Venn Diagram by putting a box around the whole thing:
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Introduction to Set Theory
Now you can see ALL your ten colleagues, neatly sorted into what sport they play (or not!). And then we can do interesting things like take the whole set and subtract the ones who play Soccer:
We write it this way: U−S = {blair, erin, francis, glen, ira, jade} Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}" In other words "everyone who does not play Soccer".
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Introduction to Set Theory
COMPLEMENT: And there is a special way of saying "everything that is not", and it is called “complement". It is denoted by:
S' or Sc
Which means "everything that is NOT in S", like this:
Sc = {blair, erin, francis, glen, ira, jade} (just like the U − C example from above)
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Summary: •
∪ is Union: is in either set
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∩ is Intersection: must be in both sets
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− is Difference: in one set but not the other
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Ac is the Complement of A: everything that is not in A
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Empty Set: the set with no elements. Shown by {}
•
Universal Set: all things we are interested in
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Introduction to Set Theory
SOLVED PROBLEMS: 1. At a breakfast buffet, 93 people chose coffee and 47 people chose tea. 25 people chose both coffee and tea. If each person chose at least one of these beverages, how many people visited the buffet? Sol: Suppose A = no. of people chose coffee =93 B = no. of people chose tea =47 A ∊ B = no. of people who chose both coffee and tea=25 Only coffee= 93-25=6 Only tea= 47-25=22 If each person chose at least one out of these two, then total no of people =68+22+25 =115 visited the buffet.
2.
Sol:
In a class of 30 students, 19 are studying French, 12 are studying Spanish and 7 are studying both French and Spanish. How many students are not taking any foreign languages? Total students=30, Studying French ,A=19, Studying Spanish, B=12, Studying both french and spanish (green) =7. Only french=19-7=12 Only spanish=12-7=5 Total students studying foreign language=12+5+7=24 So, students not taking any foreign languages=30-24=6
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Introduction to Set Theory
EXERCISE: 1 .If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∪ B? 2. If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∩ B? 3. If X = {a, e, i, o, u} and Y = {a, b, c, d, e}, then what is Y - X ? 4. In the Venn diagram, what is the set U - T?
5. The Universal Set = { x ∈ Z | - 4 ≤ x < 4} and A = {0}. What is the complement of A? 6.
If, P = The set of whole numbers less than 5 Q = The set of even numbers greater than 3 but less than 9 R = The set of factors of 6 Then what is (P ∩ Q) ∪ (Q ∩ R)?
7.
If, P = The set of whole numbers less than 5 Q = The set of even numbers greater than 3 but less than 9 R = The set of factors of 6 Then what is (P ∪ Q) ∩ (Q ∪ R)?
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Introduction to Set Theory
8.
From the above Venn diagram, what is the set (S â&#x2C6;Š T) â&#x2C6;Š V? 9.
If, A is the set of factors of 15, B is the set of prime numbers less than 10 C is the set of even numbers less than 9, then what is
10. If A, B, and C are any three sets, then which of the following is equal to Hint: Use Venn diagrams to help you answer this question. 11.
100 students were interviewed 28 took PE , 31 took BIO , 42 took ENG, 9 took PE and BIO, 10 took PE and ENG, 6 took BIO and ENG, 4 took all three subjects. a) How many students took none of the three subjects? b) How many students took PE but not BIO or ENG? c) How many students took BIO and PE but not ENG?
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Introduction to Set Theory
Notes:-
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Introduction to Set Theory
Scottish Church College B.ED Department (2013-14) e-book https://issuu.com/madhu.2su/docs/settheoryforebook.docx/15?e=0
MADHURIMA CHANDRA ROLL NO- T-031 METHOD I â&#x20AC;&#x201C; MATHEMATICS
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