Discrete Math Sets

2. What is Sets?
A set is any well-defined collection of
objects, called the elements or members of
the set.
Sets are used to group objects together.
Often, but not always, the objects in a set
have similar properties. In computer science,
sets are basic data structures

3. A set is an unordered collection of objects,
called elements or members of the set. A set
is said to contain its elements.
We write a ∈ A to denote that a is an
element of the set A. The notation a ∉ A
denotes that a is not an element of the set
A.

4. 1. Sets symbol
 Sets are denoted by using uppercase letters
for example A, B, C, D.
 Elements of sets are denoted by using
lowercase letters for example a, b, c, d or 1,
2, 3, 4.
 Groups of sets are denoted by using bracket
"{" and "}", for example { … }, { … }.
 Each elements of sets is separated by using
comma ",", for example { 1, 2, 3, 4 } … .
2. Example of sets A = { 1, 2, 3, 4 } or B = { a, b, c,
d, e }

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8.  N: Sets of Natural Numbers (Counting numbers).
objects. Natural numbers do not include 0 or negative
numbers. Ex. 1,2,3,4…….. ∞
 Z: Sets of all Integers. Whole number (not a fractional
number) that can be positive, negative, or zero.
 Z +
: Sets of all positive Integers
 Z -
: Sets of all negative Integers

9.  Q: Sets of all Rational Numbers. Any number that can be
written as a ratio (or fraction) of two integers is a
rational number.
 R: Sets of Real Numbers. The real numbers include
natural numbers or counting numbers, whole numbers,
integers, rational numbers (fractions and repeating or
terminating decimals), and irrational numbers.

10. B
B

11. The empty set There is a special set that
has no elements. This set is called the
empty set, or null set, and is denoted
by ∅ .
The empty set can also be denoted by { }.
Often, a set of elements with certain
properties turns out to be the null set.
A: Set of prime numbers between 24 and
28
B: { }

13. 1. Write set V which is represented letters between C and J in
the English Alphabet.
2. Write set O which is represented odd positive integers less
than 10.
3. Write set E which is represented even integers greater than
10 and less than 20.
4. Write set A which is represented positive integers greater
than 95 and equal to 100
5. Write set F which is represented positive odd numbers
between 5 and 7.

14. Venn diagrams
Sets can be represented graphically using Venn diagrams,
named after the English mathematician John Venn, who
introduced their use in 1881. In Venn diagrams the
universal set U, which contains all the objects under
consideration, is represented by a rectangle. (Note that the
universal set varies depending on which objects are of
interest .)

15. Inside this rectangle, circles or other geometrical figures
are used to represent sets. Sometimes points are used to
represent the particular elements of the set. Venn diagrams
are often used to indicate the relationships between sets. We
show how a Venn diagram can be used in an example.

16. A ={1,2,3,4,5,6,7}
B ={6,7,8,9,10,11}

18. Union
The union of two sets is a set containing all elements that are in A or
in B (possibly both).

19. Commutative Law: The order of the sets in which the operations
are done, does not change the result.

20. Associative Law: The grouping of two or more sets performing any operation
does not affect the next set of grouping.

21. Identity Law: In logic, the law of identity states that each thing is identical with
itself.

22. Idempotent: The union of any set A with itself gives the set A. Is the property of
certain operations in mathematics that can be applied multiple times without
changing the result. Intersection and union of any set with itself revert the same set
A U A = A

23. Law of U: Universal set is a set which has elements of all the related
sets, without any repetition of elements.

25. Commutative Law: The order of the sets in which the operations
are done, does not change the result.

26. Associative Law: The grouping of two or more sets performing any operation
does not affect the next set of grouping.

27. Properties of Intersection of Sets
Idempotent: The union of any set A with itself gives the set A. Is the property of
certain operations in mathematics that can be applied multiple times without changing
the result. Intersection and union of any set with itself revert the same set.

28. Law of U: Universal set is a set which has elements of all the related
sets, without any repetition of elements.

34. Algebra of Sets
Sets under the operations of union, intersection, and
complement satisfy various laws (identities) which are listed in Table 1.
Idempotent Laws (a) A ∪ A = A (b) A ∩ A = A
Associative Laws (a) (A ∪ B) ∪ C = A ∪ (B ∪ C) (b) (A ∩ B) ∩ C = A ∩ (B ∩ C)
Commutative Laws (a) A ∪ B = B ∪ A (b) A ∩ B = B ∩ A
Distributive Laws (a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (b) A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)
De Morgan's Laws (a) (A ∪B)
c
=A
c
∩ B
c
(b) (A ∩B)
c
=A
c
∪ B
c
Identity Laws
(a) A ∪ ∅ = A
(b) A ∪ U = U
(c) A ∩ U =A
(d) A ∩ ∅ = ∅
Complement Laws
(a) A ∪ A
c
= U
(b) A ∩ A
c
= ∅
(c) U
c
= ∅
(d) ∅
c
= U
Involution Law (a) (A
c
)
c
= A