MATHEMATICAL REASONING

MATHEMATICAL LOGIC

LOGIC IS THE SUBJECT WHICH DEALS WITH THE PRINCIPLE OF REASONING.

Logic is sometimes defines as science of proof.Mathematics and other science subjects deals with the reasoning and the arguments and every student of mathematics and other science should know the principles of logic.

Statement

A sentence is called statement if it is either true or false but not both.

Example-"two plus five is equal to seven"" is a statement because the sentence true.

A true statement is called valid statement.

A false statement is called invalid statement.

Properties of Statements

A statement is a mathematical acceptable sentence
Mathematical statements are either true or false, they cannot be both
Any exclamatory sentences are not mathematical statements
An imperative sentence is not a valid mathematical statement
An interrogative sentence is not a mathematical valid statement
A sentence involving variables such as here, there, today, tomorrow, and yesterday or any pronouns is not a statement.
Ambiguous sentences are invalid in mathematics

Types of sentences

Assertive sentence - A sentence that makes an assertion is called an assertive sentence or a declarative sentence. Example- The sun is star.

Imperative sentence - A sentence that expresses a request or a command is called Imperative sentence. Example- Give me a glass of water.

Exclamatory sentence- A sentence that express some strong feeling is called exclamatory sentence. Example -May I help you.

Interrogative sentence - Aseentence that ask some question called interrogative sentence. Example- How young are you?

Optative sentence - A sentence that express a wise is called optative sentence. Example- God bless you.

Logical statement or Proposition - A logical statement is any statement which is either true or false is equivalently valid or invalid.The falseness or truth of a statement is calleed its truth value.An open statement is not a statement. Example- 1. "Good morning to all " is not a statement. 2. Every rectangle is a square" is a false statement.

Open statement - A sentence which contains one or more variable such that when certain values are given to the variable it becomes a statement, is called an open statement.

Compound statement - If two or more simple statements are combined by the use of words such as "and","or","not","if",""then","if and only if", then the resulting statement is called a compound statement.

Negation of Statements

The denial of the statements is basically called the negation of the statements. It is denoted by the sign (∼). It is read as ‘not’. If p: Number 7 is an odd number. The negation of p is not p, ∼ p: Number 7 is not an odd number.

Truth value and Truth table

Truth value - A statement can be either 'true' or 'false' are called the truth values of the statement and it is represented by the symbol T,F respectively.

Truth table- A truth table is a summary of truth values opf the resulting statements for all possible assignment of \values to the variables appearing in a comound statement.Number of rows depends on their number of statements.

1. Truth table for single statement p
number of rows = 21 = 2

2. Truth table for two statements p and q
number of rows = 2² = 4

A convenient and helpful way to organize truth values of various statements is in a truth table. A truth table is a table whose columns are statements, and whose rows are possible scenarios. The table contains every possible scenario and the truth values that would occur.

Elementary operations on logic

connectives - The words 'and,'but','not','either....or','neither....nor','if,'then,'.....if and only if.....' etc .., are called connectives.

The Three Types of Connectives are:

Conjunction: A conjunctive statement is the compound statement that is obtained by connecting two simple statements with the connective ‘and’. The conjunction of two statements ‘p’ and ‘q’ is given as “p and q” or “p ∧ q”. Example: The conjunction of “Time is money” and “Money is time” is given as “Time is money and money is time”.

Disjunction: A disjunctive statement is the compound statement that is obtained by joining two or more statements with a connecting word ‘or’. The disjunction of two statements ‘p’ and ‘q’ is given as “p or q” or “p ∨ q”. Example: The disjunction of two statements “A natural number is an odd number” and “A natural number is an even number” is given as “A natural number is an odd number or an even number”.
Negation: Negation of a statement is the statement representing denial of any statement. The word ‘not’ is used to negate a statement. Through negation does not join two statements, it gives the negative of a statement. Negation of any statement ‘p’ is given as “not p” and is symbolically represented as “~p”.

Example: Consider a statement “7 is a prime number”. The negation of this statement is given as “7 is not a prime number”.

Laws of Algebra of Statements

Idempotent Laws

If a is any statement then
a ∧ a = a
a ∨ a = a

Associative Laws

Consider a, b, and c be the three statements, then
(a ∧ b) ∧ c = a ∧ (b ∧ c)
(a ∨ b) ∨ c = a ∨ (b ∨ c)

Commutative Laws
If a and b are the two statements, then
a ∧ b = b ∧ a
a ∨ b = b ∨ a

Distributive Laws

Consider a, b, and c be the three statements, then
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

De Morgan’s Laws

For the statements a and b
∼ (a ∧ b) = (∼a) ∨ (∼b)
∼ (a ∨ b) = (∼a) ∧ (∼b)

Identity Laws

For any statement a, then
a ∧ F = F
a ∧ T = a
a ∨ T = T
a ∨ F = a

Complement Laws
For any statement a,a ∧ (∼a) = Fa ∨ (∼ a) = T∼(∼a) = a∼T = F∼F = T

Solved Example for You

Problem: Write the contrapositive and the converse of the following statements.

If n is an odd number then n2 is odd.
If a is square then all its four sides are equal.

Solution: Contrapositive statements

If n2 is not an odd number, it is not an odd number.
If all the sides are not equal, a is not square.

Converse statements

If n2 is odd, then n is odd.
If all the four sides of a are equal, then it is a square.

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