1. Examples
1. Some definitions
2. Conjuction
The compound statement obtained by combining two simple statements by the connective 'and' is called conjunction of these simple statements.
The conjunction of `p`, `q` is denoted by `p ^^ q` and read as `p` and `q`
The compound statement `p ^^ q` is true only when `p` and `q` are both true and in all other cases it is false.
The truth table for `p ^^ q` is given below
3. disjuction
The compound statement obtained by combining two simple statements by the connective 'or' is called disjuction of these simple statements.
The disjuction of `p`, `q` is denoted by `p vv q` and read as `p` or `q`
The compound statement `p vv q` is false only when `p` and `q` are both false and in all other cases it is true.
The truth table for `p vv q` is given below
4. Negation
A statement whose truth value is opposite to that of a given statement is called negation.
The negation of `p` is denoted by `~p`
The truth table for `~p` is given below
5. Logically equivalent statement
If two statements `S_1` and `S_2` have the same truth value for all possible truth values of the statements, they are said to be logically equivalent statements.
`S_1=S_2`
6. Tautology
A statement which is always true is called tautology. It is denoted by `t`
`p vv t=t`
`p ^^ t=p`
`p vv (~p)=t`
7. Contradiction
A statement which is always true is called contradiction. It is denoted by `c`
`p ^^ c=c`
`p vv c=p`
`p ^^ (~p)=c`
8. Implication
A Statement of the form 'if p then q` is called an implication and is written as `p=>q` and read as p implies q. Here p is called antecedent and q is called consequent.
1. `p=>q = (~p) vv q`
2. `p=>q = (~q) => (~p)`
3. `p=>q = p ^^ (~q)`
The truth table for `p=>q` is given below
9. Double Implication
A Statement of the form 'p if and only if q` is called a double implication and is written as `p<=>q` and read as p double implies q.
Here `p<=>q` is a conjunction of `p=>q` and `q=>p`.
The truth table for `p<=>q` is given below
| `p` | `q` |
`p=>q` |
`q=>p` |
`p<=>q` | | T | T |
T |
T |
T | | T | F |
F |
T |
F | | F | T |
T |
F |
F | | F | F |
T |
T |
T |
This material is intended as a summary. Use your textbook for detail explanation.
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