Discrete
Mathematics - 2
UNIT-1: - SET, RELATIONS AND FUNCTIONS
SET: - A set is a well-defined collection of distinct objects.
Examples of sets are : -
The group of all sophomores in
the university is a set. The group of all Computer Science - majors in the
university is a set, The group of all second-year Computer Science students are
a set.
NOTATION:-
We use the notation {a,
b, c} to denote the set which is the collection of the objects a, b, and c. The
objects in a set are also called the elements or the members of the set. we
usually also gives the name of the set.
for examples we write M = {a, b, c} to mean that the name of this set is M is the collection of objects a, b, and c.
a ⊑ A means that objects is a number, or element, of a set A.
If A = {a, b, c} , then a ⊑ A.
Two sets are equal if they contains same elements. Note that a set contains only distinct elements, Thus , {a, b, c} is a redundant representation of the set {a, a, b, c}.
OBJECTS:- An objects could be anything. It can be something we can touch or see or it can be an idea or a concept.
Examples:- A set of all facts learned in discrete mathematics course.
A collection of pens.
A collection of cars.
A set of odd numbers divisible by 2.
A set of vowels of English Alphabets.
Q. What is not a set?
A collection of beautiful songs because this is where the well-defined come into picture.
WELL-DEFINED:- A set is considered to be well-defined if it is possible to establish that any given object belongs to the set.
Examples:- "A set of beautiful songs".
"Beautiful songs" is not well-defined . The definition of beautiful songs changes from person to person.
A song is beautiful when it is meaningful
 |
| Mizanur |
 |
| Anarul |
Therefore a set is beautiful songs is not well-defined and hence it is not set.
A set must be well-defined collection of objects.
More examples: -
- A collection of great peoples of the world.
- A set of beautiful flowers.
- A collection of best football players in the world.
- A collection of most dangerous animals found in the forest.
- A collection of the most talented boys in your class.
DISTINCT-OBJECT:-
We can have a set with duplicate objects. But a set with duplicate objects is similar to a set with distinct-objects.
For examples: - A = {1, 2, 2, 3, 3, 3}
B = {1, 2, 3}
A=B
Eventually, we end up with a set without duplicate elements , This is the reason why the definition holds true.
So, "A set is a well-defined collection of distinct object" holds true.
SET MEMBERSHIP: - Any object belonging to a set is called a member or an element of that set.
