1. Sets & Set Operations

2. Basic discrete structures
Discrete math: Study of the discrete structures used to represent discrete objects
Many discrete structures are built using sets
Sets : collection of objects
Examples of discrete structures built with the help of sets:
Combinations
Relations
Graphs

3. Set
Definition: A set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set.(Cantor's naive definition).
A naïve theory in the sense of "naïve set theory" is a non-formalized theory, that is, a theory that uses a natural language to describe sets and operations on sets.

4. Examples:
– Vowels in the English alphabet
V = { a, e, i, o, u }
– First seven prime numbers.
X = { 2, 3, 5, 7, 11, 13, 17 }

5. Representing sets Representing a set by:
1) Listing (tabulation method)) the members of the set.
2) Definition by property, using the set builder notation.(Rule method)
{x| x has property P}.
Example:
Even integers between 50 and 63.
1) E = {50, 52, 54, 56, 58, 60, 62}
2) E = {x| 50 <= x < 63, x is an even integer}

6. If enumeration of the members is hard we often use ellipses.
Example: a set of integers between 1 and 100
A= {1,2,3 …, 100}
Rational numbers
Q = {p/q | p ∈ Z, q ∈ Z, q ≠ 0}

7. Set can be
Finite set
Infinite set
Singleton set

8. A set is determined only when we know definitely what object it contains. There must be no ambiguity or doubt in this regard.
Example: The collection of all tall students in a college does not define a set.
Because there will always be some doubt about as to which boys are to be regarded as tall.
For this reason the objects of a set are required to be “well defined”.
Thus, if we speak of the collection of all students in a college who are above 170cms then we are actually speaking of set.

9. Interval Notation

10. Universal Set and Empty Set
The universal set U is the set containing everything currently under consideration.
Content depends on the context.
Sometimes explicitly stated, sometimes implicit.
The empty set is the set with no elements.
Symbolized by ∅ or { }

11. Things to remember

12. Equal sets
Definition: Two sets are equal if and only if they have the same elements.
Example:
• {1,2,3} = {3,1,2} = {1,2,1,3,2}
Note: Duplicates don't contribute anything new to a set, so remove them. The order of the elements in a set doesn't contribute anything new.

13. Subsets
Definition: A set A is said to be a subset of B if and only if every element of A is also an element of B.
We use A ⊆ B to indicate A is a subset of B.
Example:
A={1,2,3}, B={1,2,3,4,5}, C={2,3,5,6}
Clearly A ⊆ B and A ⊄ C

14. Proper Subset proper subset / strict subset A⊂B In the example
We observe that the set B which contains A as subset, contains elements that are not in A.
In such a situation we say that the set A is Properly contained in B or is a Proper subset of B.
A⊂B
proper subset / strict subset

15. Things to remember Every set is a subset of itself
Two sets A and B are equal if and only if A ⊆ B and B ⊆ A.
The Null set ∅ is a subset of every set A.
For any sets A,B,and C if A ⊆ B and B ⊆ C then A ⊆ C.
For any sets A,B,C if A=B AND B=C, then A=C.

16. Venn Diagrams
Relationship between the sets can be depicted in diagrams called Venn Diagrams.
represent A ⊆ B B A

17. For any sets A,B,and C if A ⊆ B and B ⊆ C then A ⊆ C.
The truth of the above statement is obvious from the following Venn diagram. C B A

18. The Null set ∅ is a subset of every set S

19. Power Set
Given a set S, the power set of S is the set of all subsets of S. The power set is denoted by P(S).
Example: Consider set A={a,b}
P(A)={∅, {a},{b}, {a,b}= )={∅, {a},{b}, A}
If A={1,2,3}
P(A)={∅,{1},{2},{3},{1,2},{2,3},{1,3},A}
If a finite set A has n elements, then P(A) has 2n elements.

20. If a finite set has n elements, prove that the power set of A has 2n elements
If a finite set has n elements, then subset of A can have
No element(Null set)
One element(singleton set)
Two elements(each combination of two elements of A)
Three elements(each combination of three elements of A)
…… n-1 elements and n elements(A itself)
Null set( 1 in number)
Singleton sets: n in number: nc1
Each combination of two elements of A gives a subset of A containing two elements: nc2
Each combination of 3 elements of A gives a subset of A containing 3 elements : nc3
..... ncn-1
n elements… ncn
Binomial theorem: (1+x)n
1+nc1+nc2+nc3……………+ncn-1+ncn= (1+1)n= 2n

21. Cardinality
|{∅}| = 1

22. Set Operators: Union
Definition: The union of two sets A and B is the set that contains all elements in A, B, r both. We write:
AB = { x | (a  A)  (b  B) } U A B

23. Set Operators: Intersection
Definition: The intersection of two sets A and B is the set that contains all elements that are element of both A and B. We write:
A  B = { x | (a  A)  (b  B) } U A B

24. Disjoint Sets
Definition: Two sets are said to be disjoint if their intersection is the empty set: A  B =  U A B

25. Set Difference
Definition: The difference of two sets A and B, denoted A\B or A−B, is the set containing those elements that are in A but not in B U A B

26. Set Complement
Definition: The complement of a set A, denoted A ($\bar$), consists of all elements not in A. That is the difference of the universal set and A: U\A
A= AC = {x | x  A } U A A

27. Set Complement: Absolute & Relative
The (absolute) complement of A is A=U\A
The (relative) complement of A in B is B\A U U B A A A

28. Relative Compliment
Given two sets A and B, the set of all the elements that belong to B but not A is called the compliment of A relative to B(or relative compliment of A in B) and is denoted as B-A
B-A={x|x∈B and x ∉ A}.
The set A-B(Relative compliment of B in A) can be defined similarly.
B-A and A-B are not same

29. Things to remember
For any sets A and B, the sets A-B and B-A are disjoint.
If A and B are disjoint, then A-B=A, B-A=B.
If U is a universal set and A ⊆ U, the U-A= Ac
A=(A U B)-(B-A), B=(AUB)-(A-B)
A=(A ∩ B)U(A-B),B=(A ∩ B)U(B-A)

30. Symmetric Difference
For Two sets A and B, the releative compliment of A ∩ B in A U B is called symmetric difference of A and B denoted as A∆B or A⊖B.
Simply A∆B= (A – B) ∪ (B – A)
A⊖B=(A U B)-(A ∩ B )
={x|x ∈ A U B and x ∉ A ∩ B }

31. Set Identities Identity laws A ∪ ∅ = A A ∪ U = U A ∪ A = A A ∩ U = A
Domination laws
Idempotent laws
Complementation law
(A) = A
Complement laws
A ∩ A = ∅ A ∪ A = U

32. Set Identities (cont.) Commutative laws A ∪ B = B ∪ A Associative laws
A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C
Distributive laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Absorption laws
A ∪ (A ∩ B) = A
De Morgan’s laws
A ∪ B = A ∩ B
A ∩ (A ∪ B) = A
A ∩ B = A ∪ B

33. Duality Each union replaced by intersection
Each intersection replaced by union
U by ∅ and ∅ by U
If two sets P and Q are equal then their duals also equal.

34. Cartesian Product of sets
Definition: Let A and B be two sets. The Cartesian product of A and B, denoted AxB, is the set of all ordered pairs (a,b) where aA and bB
AxB = { (a,b) | (aA)  (b  B) }
The Cartesian product is also known as the cross product

35. A1A2… An ={ (a1,a2,…,an) | ai  Ai for i=1,2,…,n}
Cartesian Product
Cartesian Products can be generalized for any n-tuple
Definition: The Cartesian product of n sets, A1,A2, …, An, denoted A1A2… An, is
A1A2… An ={ (a1,a2,…,an) | ai  Ai for i=1,2,…,n}