Mathematics is the study of numbers and shapes with functions and quantities - well this is tough to properly define mathematics though. But in the subliminal level, we know what mathematics is :). Mathematics is the language and the representation of the universe.

Mathematics can be categorized into continuous and discrete. Continuous mathematics deals with study of the behaviour of real world and nature like planetary motions, time, electricity, etc.

For the computer science perspective discrete mathematics is important. Discrete mathematics is the study of things that are countable and can be separated in their distinct values. Discrete values can be colors, alphabets and other values like these. In case of computers, The fundamental discrete values are - bits, the 0s and the 1s. So, Computers operate in discrete steps and store data as discrete values.

Some topics to cover in discrete mathematics

This list is irrespecive of the order:

• Sets
• Relations
• Functions
• Monoids
• Graphs
• Combinatorics
• Partial orders
• Lattice

Let us go through the very basics of these topics in this post and we will cover these topics in detail in other posts later. But for now, let's just get started by defining what are Statements

Statements

A statement is a sentence which is either true or false. There are two types of statements:

• Atomic: If a statement can't be divided into further small statements then it is called an atomic statement.
• Molecular: If a statement can be divided into smaller statmenets then they are called molecular statement.

A statement is just a sentence that is either true or false. But this is a fundamental concept of discrete mathematics or even the maths in general.

Here are the following examples of a statement:

1. A human has two legs.
2. The Earth is rectangular.
3. 5 + 6 = 11
4. There are 3 bridges constructed between Earth and Moon.

The above are some statements examples which are either true or false. Point 1 and 3 are true and point 2 and 4 are false.

Following are some examples of non-statements:

1. May I go outside to play?
2. x + 5 = 21
3. Give me a glass of water
4. The blue flowers has

Here, point 1 is straightway a question which is not giving any implication. This is just a question and you can't tell if it is true or false sentence.

Point 2 is an expression where we have "x" and we don't know the value of x - it is a variable based on that it can be true or false but we don't know until we know the value of x.

Point 3 is like and order that someone is giving to someone.

Point 4 is just an incomplete sentence so we can't say anything about it, it also doesn't imply anything.

None of these 4 points are giving us any conclusion of whether they are true or false and so they are not statements.

Now, the above valid statements were atomic statements as there is no way we can break them into more smaller statements. Let us now look at the molecular statements. Consider the following statement:

81 is a multiple of 9 and cats live inside water.

This statement is a molecular statement. There are two atomic statements in it:

1. 81 is a multiple of 9
2. cats live inside water.

These two statements are connected by the "and" word - these connectors are called logical connectives. Following are the logical connectives.

1. and (∧)
2. or (∨)
3. if...then... (→)
4. if and only if (↔)
5. not (¬)

1 to 4 are binary connectives because they can be used for connecting two statements. The 5th one, "not" is unary statement because it is used with single statement.

For molecular statements, to derive whether they are true or false, they can be derived from following way:

1. The molecular statements are made up of atomic statements. So they are either true or false, so we can say taht they are boolean values.
2. The logical connectives are just the boolean expressions, the boolean operations can be applied on these boolean values, i.e., the atomic statements.

If we take this example:

Let,
s = 81 is the multiple of 9 and cats live inside water
We can deduce it as:
let,
a = 81 is the multiple of 9 = true
b = cats live inside water = false
So, the molecular statement

S = a and b
S = a ∧ b
S = true ∧ false

So,
S = false
That means, this molecular statement is false

This was about the very basics of discrete mathematics and other topics will be covered in other posts.