Sets and subsets

D. K. Bhattacharya

Set

• It is just things grouped together with a certain property in common. Formally it is defined as a collection of well defined objects, so that given an object we should be able to say whether it is a member of the set or not.

• For example, the items we wear: these would include shoes, socks, hat, shirt, pants, and so on.

• Obviously if somebody asks whether a ‘leg’ belongs to this collection, the answer is ‘no’. But if it is asked whether a ‘vest’ belongs to this collection, the answer is ‘yes’.

• Another example is types of fingers of the left hand: these are index, middle, ring, and pinky.

• The third example is all 26 English letters: these are a, b, c, ..

Notation

• There is a fairly simple notation for sets. We simply list each element, separated by a comma, and then put some curly brackets around the whole thing.

• Notations for the above sets are the following:• {socks, shoes, watches, shirts, ...}

{index, middle, ring, pinky}• {a, b, c, ..., x, y, z}

Numerical Sets

• Set of even numbers: {..., -4, -2, 0, 2, 4, ...}(infinite set)

• Set of odd numbers: {..., -3, -1, 1, 3, ...}(infinite set)

• Set of prime numbers: {2, 3, 5, 7, 11, 13, , ...}(infinite set)

• Positive multiples of 3 that are less than 10: {3, 6, 9} (finite set)

Some standard notations

• Usually a set is denoted by a capital letter, but an element of the set is denoted by a small letter. Thus ‘A’ denotes a set, but ‘a’ denotes an element of the set A.

• If a belongs to A, we denote it by the symbol

• If a does not belong to A, we denote it by the symbol

a A

a A

Equality of sets

• Equality of two subsets:• If A and B are two sets such that every element of

A is an element of B, and every element of B is an element of A, then the two sets A and B are equal. It is denoted by A = B.

• Example:• A is the set whose members are the first four

positive whole numbers and • B = {4, 2, 1, 3}

Proper and improper subsets

• A set A is called a proper subset of B if every element of A is an element of B, but A is not whole of B. It is denoted by .

• Similarly a set B is called a proper subset of A if every element of B is an element of A, but B is not whole of A. It is denoted by

A B

.B A

• A set A is called an improper subset of B if every element of A is an element of B, where A may be whole of B. It is denoted by

• In particular, A is an improper subset of A. It is denoted by

• If A is a subset of B, then B is called a superset of A. is called ‘inclusion’ and is called ‘containing’. Thus A is included in B, but B contains A.

A B

A A

Examples

• 1. Is A a subset of B?, or B a subset of A, where

• A = {1, 3, 4} and B = {1, 4, 3, 2}?• 2. 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?

• A = {..., -8, -4, 0, 4, 8, ...}• B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}

Universal set and null set

• If we consider infinite number of subsets

• Then to make the process meaningful, we are to consider existence of a set U such that

• The set U is called the universal set. In fact, all sets are contained in U.

1 2 3 4 ...A A A A

1 2 3 4 .....A A A A U

• If we consider infinite number of subsets

• Then to make the process meaningful, we are to consider existence of a set such that

• The set is called a null set or an empty set. In fact, it is contained in every set.

1 2 3 4 ...A A A A

1 2 3 4 ...A A A A

Power set

• Power set of a set A is the collection of all its subsets.

• Example: Power set of {1,2,3}• These are three point set {1,2,3}, one in number;

two point sets {1,2}, {1,3}, {2,3}, three in number; one point set {1}, {2}, {3}, three in number and the single null set

• Total number of subsets is . In general the power set of a set of n elements is

38 2

2 .n

Example of voting primaries

OPERATIONS ON SETS

• Sets can be combined in a number of different ways to produce another set. Here four basic operations are introduced.Definition (Union): The union of sets A and B, denoted by A B , is the set defined as

A B = { x | x A or x B or to both}

• Example 1: If A = {1, 2, 3} and B = {4, 5} • then A B = {1, 2, 3, 4, 5} .

Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5}, • then A B = {1, 2, 3, 4, 5} .

• Definition (Intersection): The intersection of

• sets A and B, denoted by A B , is the set

• defined as A B = { x | x A and x B }

• Example 3: If A = {1, 2, 3} and B = {1, 2, 4, 5}, then A B = {1, 2} .

Example 4: If A = {1, 2, 3} and B = {4, 5}, then A B = .

• Definition (Difference): The difference of sets A from B , denoted by A - B , is the set defined as

A - B = { x | x A and x B }

• Example 5: If A = {1, 2, 3} and B = {1, 2, 4, 5}, then A - B = {3} . Example 6: If A = {1, 2, 3} and B = {4, 5}, then A - B = {1, 2, 3} .

Note that in general A - B B - A

• Definition (Complement): For a set A, the difference U - A , where U is the universal set, is called the complement of A and it is denoted by . It is the set of everything that is not in A.

A or Ac

VENN DIAGRAM

Complement of P

P-Q

Example of P-Q

Solve the following problems with the use of a Venn diagram.

1. In a class of 50 students, 18 take Chorus, 26 take Band, and 2 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band? [Ans. 8]

16 + 2 + 24 + x = 5042 + x = 50x = 8 students

2. In a school of 320 students, 85 students are in the band, 200 students are on sports teams, and 60 students participate in both activities. How many students are involved in either band or sports? [Ans. 225]

25 + 60 + 140 = 225There are 225 students involvedin either band or sports.

3. A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all three kinds of pets, how many patrons have none of these pets? [Ans.5]

7 + 4 + 0 + 3 + 1 + 5 + 1 + x = 2621 + x = 26x = 5 patrons have none of these animals

4. A guidance counselor is planning schedules for 30 students. Sixteen students say they want to take French, 16 want to take Spanish, and 11 want to take Latin. Five say they want to take both French and Latin, and of these, 3 wanted to take Spanish as well. Five want only Latin, and 8 want only Spanish. How many students want French only? [Ans.7]

x + 4 + 3 + 2 + 5 + 8 + 1 = 30x + 23 = 30x = 7

More examples of sets and Venn diagrams

• Ten Best Friends• You could have a set made up of your ten best friends:• {alex, blair, casey, drew, erin, francis, glen, hunter, ira,

jade} • Each friend is an "element" (or "member") of the set • 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).

And casey, drew and jade play Tennis:Tennis = {casey, drew, jade}Soccer Tennis = {alex, casey, drew, hunter, ∪

jade}

Soccer ∩ Tennis = {casey, drew}

• Soccer − Tennis = {alex, hunter}

• Volleyball = {drew, glen, jade}• S means the set of Soccer players• T means the set of Tennis players• V means the set of Volleyball players

Drew plays Soccer, Tennis and Volleyball jade playsTennis and Volleyball alex and hunter play Soccer, but don't play Tennis or Volleyball no-one plays only Tennis

• S = {alex, casey, drew, hunter}• T ∪ V = {casey, drew, jade, glen}• S ∩ V = {drew}• (S ∩ V) − T = { }