This post deals with propositional calculus. In propositional calculus we deal with propositions. Propositions are those sentences which are either true of false. One and the most important thing which is required in propositional calculus is reasoning. Our ancestors were able to form civilizations as they were able to reason things. But a rigorous study of logical reasoning was not done for a long time. The first such study that has been found is by the Greek philosopher Aristotle (384-322 BC). Leibnitz (1646-1716) and George Boole (1815-1864) seriously studied this and came up with with a theory and called it symbolic logic.
'The sun rises in the East.' is a proposition. It is a proposition because it can only take two values either true or false. The sentences which are either true or false are propositions or statements. In mathematics we come across many propositions and statements. x>2 is not a proposition as it can be true or false according as x>2 or x<2. So, 2<3 is a proposition but x<2 is not a proposition.
There are three main connectives in propositional calculus. It is conjunction(∧), disjunction(∨) and negation(¬).
Consider the two statements.
p: I play badminton.
q: I play football.
We can frame two statements with the combination of A and B. It is called compound statement.
I play badminton and I play football.
The above statement is formed by the combination of the two statements with the help of 'and'. This is called conjunction. It is denoted by the symbol '∧'. We can frame a table of all the possibilities. It is called truth table.
As we can see from the above that conjunction is true only when all the possibilities are true and false if anyone is false.
I play badminton or I play football.
The above statement is formed by the combination of the two statements with the help of 'or'. This is called disjunction. It is denoted by the symbol '∨' We can frame a truth table of all the possibilities.
As we can see from the above that conjunction is false only when all the possibilities are false and true if anyone is true.
I don't play football.
The statement 'I play football.' is transformed to 'I don't play football.' with the help of 'negation'. It is denoted by '¬'. The truth table for 'negation' is
Negation of q is true when q is false and false when q is true.
'The sun rises in the East.' is a proposition. It is a proposition because it can only take two values either true or false. The sentences which are either true or false are propositions or statements. In mathematics we come across many propositions and statements. x>2 is not a proposition as it can be true or false according as x>2 or x<2. So, 2<3 is a proposition but x<2 is not a proposition.
There are three main connectives in propositional calculus. It is conjunction(∧), disjunction(∨) and negation(¬).
Consider the two statements.
p: I play badminton.
q: I play football.
We can frame two statements with the combination of A and B. It is called compound statement.
I play badminton and I play football.
The above statement is formed by the combination of the two statements with the help of 'and'. This is called conjunction. It is denoted by the symbol '∧'. We can frame a table of all the possibilities. It is called truth table.
| F is false. T is true. | |||
|---|---|---|---|
| p | q | p∧q | |
| F | F | F | |
| F | T | F | |
| T | F | F | |
| T | T | T | |
As we can see from the above that conjunction is true only when all the possibilities are true and false if anyone is false.
I play badminton or I play football.
The above statement is formed by the combination of the two statements with the help of 'or'. This is called disjunction. It is denoted by the symbol '∨' We can frame a truth table of all the possibilities.
| F is false. T is true. | |||
|---|---|---|---|
| p | q | p∨q | |
| F | F | F | |
| F | T | T | |
| T | F | T | |
| T | T | T | |
As we can see from the above that conjunction is false only when all the possibilities are false and true if anyone is true.
I don't play football.
The statement 'I play football.' is transformed to 'I don't play football.' with the help of 'negation'. It is denoted by '¬'. The truth table for 'negation' is
| F is false. T is true. | |||
|---|---|---|---|
| q | ¬q | ||
| F | T | ||
| T | F | ||
Negation of q is true when q is false and false when q is true.
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