Compound statements `p` and `q` are combined by the connective `'OR' (∨)`, then the compound statement denoted as `p ∨ q` (`p` or `q`) so formed is called a `text(disjunction.)`
`e.g.` Two lines in a plane either intersect at one point or they are parallel.
Sometimes we use the connective 'either .. or .. 'to obtain `p ∨ q` and read `p ∨ q` as 'either `p` or `q`'.


`text(Note :)`
`(i)` A compound statement with an ` 'Or'` is true when one component statement is true or both the component statement are true.
`(ii)` A compound statement with an `'Or'` is false when both the component statements are false.

`text(Imp.: e.g.)` A student who has taken biology or chemistry can apply for M.Sc. microbiology program.
This means that student who have taken both biology and chemistry or only biology or only chemistrycan apply for the microbiology program. This is example of inclusive "Or". In this case truth table is same as `p ∨ q`.

`text(Imp.: e.g.)` Student can take French or Sanskrit as their third language.
This means that student have to choose only one subject from French and Sanskrit. It exclude the case when one student can choose both subject. This is case of exclusive "Or". This is represented as `p ⊻ q` or `p otimes q`. Truth table for `text(exclusive or)` is as follows.

The denial of a statement is called the negation of the statement denoted as `~` .
e.g. `p : ` Everyone in Germany speaks German.
`~ p: ` it is false that everyone in Germany speaks German.
While forming the negation of a statement, phrases like, "It is not the case" or "It is false that" are also used.

e.g. `p : `All integers are rational numbers".
` ~p:` Atleast one integer is not a rational number.

If `p` is true then `~ p` must be false and if `p` is false then `~ p` must be true


It maybe noticed that `~ (~ p)= p`. Also `p` and `~ p` are contrary.
e.g. the statements ` 'x `is an even number' and `' x` is an odd number' are contrary if `x` is a whole number because both the statements cannot have the same truth value.

`text(Imp. :)` If may be observed that negation is not a binary operation, it is a unary operation i.e. a modifier.
1. ` ~ p` is true iff `p` is false.
2. `~ p` is false iff `p` is true.

Quantifiers are phrases like `text("There exists")` and `text("for all". )`

`text(Negation of Quantifiers)`

`(i)` `P =` There exist a number which is equal to its square .
`\ \ \ \ \ \ ~ P =` There does not exist a number which is not equal to its square.

`(ii) P =` For every real number `x, x` is less than `x + 1`.
`\ \ \ \ \ \ \ ~ P =` There exist a number for which `x` is not less than `x + 1`.


`text(Implication :)`

There are three types of implications:
`(i)text( "If ..... then")`
`(ii)text( "Only if")`
`(iii)text( "lf and only if")`

`( 1) text( "If ..... then")` type of compound statement is called `text(conditional statement.)`

The statement `text('if p then q') ` is denoted by `p-> q` (to be read as `text('p implies q')`) or by `p => q`. `text(Note that)` `p -> q` also means
(i) `p` is sufficient for `q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)` `q` is necessary for `p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iii)` `p` only if `q`
(iv) `p` lead to `q\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (v)` `q` if `p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (vi)` `q` when `p`
(vii) if `p`, then `q`


e.g. `p : a` number is a multiple of `9`
`\ \ \ \ \ \ \ q : a` number is a multiple of `3`.
Then `p -> q` or `p => q`


`p-> q` is false only when `p` is true and `q` is false. Truth table for `p-> q` is as follows.

`(2)text( "If and only if")` type of compound statement is called `text(Biconditional or equivalence or 'double implication')`. Symbolically `text('p iff q')` is represented by `p <-> q` or by `p <=> q`.
(i) `p` is a necessary and sufficient condition for `q`.
(ii) `q` is necessary and sufficient condition for `p`.
(iii) lf `p` then `q` and if `q` then `p`
(iv) `q` if and only if `p`.
e.g. `p:` If the sum of digits of a number is divisible by `3`, then the number is divisible by `3`.
`\ \ \ \ q :` if a number is divisible by `3`, then the sum of its digits is divisible by `3`.


A number is divisible by `3` if and only if the sum of its digits is divisible by `3`.
The following are other illustrations which actually do not appear to be so but they in fact are biconditional.
`(i)` If you work hard only then you can succeed.
`(ii)` You can go on leave only if your boss permits. The truth table for biconditional is as follows: