Mathematical Logical Statements | Readstall

Mathematical Logical Statements

05-12-18 Arjun Y 0 comment

Statements:

In this way to do mathematics, we can use statements. By using statements, we must be able to talk and write about mathematics. Statement is any declarative sentence, which is either true or false.

*In this we have two types & statements:

  1. Atomic statements
  2. Molecular statements

 

Atomic Statement:
A statement, which is can not be divided into smaller statements. It is called as atomic statements.

Ex: This table is big.

Molecular Statement:
A statement which is can be divided into smaller (or) sub statements. It is called as molecular statements.
Ex: 1 + 3 + 5 + 7—–+2n – 1

In this we can build more complicated molecular statements by combining of simple atomic statements using connectives are known as these are used to connect one (or) more statements and can form single statements. We can have 5 types connections.

They are:

  1. Conjunction
  2. Dis-junction
  3. Negation
  4. Conditional
  5. Bio-Conditional

Conjunction:
Conjunction means performing and operation between two (or) more statements. 
Logical Symbol = ∧ called as “AND”, in this operation, when both statements are true then we have conjunction values is true other wise false.

Ex:

  • Ramu is a good boy.
  • Ramu studying 10th class

Conjunction:
Ramu is good boy AND Ramu is studying 10th class.

Truth Table for Conjunction:

P Q P∧Q
T T T
T F F
F T F
F F F

Disjunction: Disjunction means performing OR operation between (or) more statements.Logical symbol = V called as “OR” In this operation, X when one & the statement is true then we have conjunction value is true, otherwise false. When both statements are false then it is false.

Ex:

  1. Ramu is a good boy.
  2. Ramu studying 10th class

Dis-junction of (1) & (2) : Ramu is a good boy OR Ramu studying 10th class.

Truth Table & Dis-junction:

P Q P∨Q
T T T
T F T
F T T
F F F

Negation: Negation means performing NOT operation for one statement.Logical symbol = ~ called as “NOT” operation for one statement.

Ex:

  1. Naresh is a good boy.
  2. Negation of: Naresh is not a good boy.
  3. Naresh is not a good boy
  4. Naresh is a bad boy

Conditional:
Conditional means performing conditional operation between two (or) more statements.Logical symbol = → called as “If A then B”If P and Q are any two statements, then conditional statement “P→Q” which is read as “If P, then Q”.In the P→Q conditional statement, when P value is true and Q value is F then P→Q has false value, other wise it is true.

Ex:

  1. Ramu is a good boy
  2. Ramu studying 10th class

Conditional & (1) & (2) ⇒ If Ramu is good boy then Ramu is studying 10th class.

Truth Table:

P Q P∨Q
T T T
T F F
F T T
F F T

Bi-Conditional:
Bi-Conditional means performing Bi-Conditional operation between two (or) more statements.
Logical symbol: ⇔ called as “P is necessary and sufficient for Q”.If P and Q are any two statements, then bi-conditional statement. “P⇔Q”, which is read as “P is necessary and sufficient for Q”.In the “P⇔Q” BI-Conditional statement, when both P&Q has same values then P⇔Q has true value Same values means both are either true or false, Other wise it is false.

Note:

  1. Q is necessary for P
  2. P is sufficient for Q
  3. Q if P
  4. P only if Q
  5. P implies Q

Truth Table:

P Q P⇔Q
T T T
T F F
F T F
F F T



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