# variants of conditional statements

Statements can have conditional relationships that are true in both directions. This is the case when a hypothesis causes the conclusion, but may be true that if the conclusion occurred first then the hypothesis could result.

This can be said for the opposite of a hypothesis and conclusion happening.

If a combination of both the converse and inverse happens, it is contrapositive.

- Converse - The concept of a hypothesis/conclusion causing one another's result in

a given direction

- Inverse - The opposite of a hypothesis and conclusion
- Contrapositive - Combination of the converse and inverse results

## Contrapositive

The contrapositive of a conditional statement `if p then q`

is obtained by both
**negating** and **exchanging the hypothesis and conclusion**.

In statement form:
`p -> q`

is `~q -> ~p`

**Note:** A conditional statement is logically equivalent to its contrapositive.
This is often easier to utilize and prove.

Example: Write a contrapositive.

"If a number is divisible by 9, then the number is divisible by 3"

`p`

= "the number is divisible by 9"
`q`

= "the number is divisible by 3"

Statement form:
`p -> q`

Contrapositive form:
`~q -> ~p`

"If the number is not divisible by 3, then it is not divisible by 9"

### Compound statements in contrapositives

A compound statement can utilize De Morgan's laws to break it down into components for determining negations.

Example: Find the contrapositive.

"If n is prime, then n is odd or n is 2"

`p`

= "n is prime"
`q`

= "n is odd"
`r`

= "n is 2"

Statement form:
`p -> (q V r)`

Equivalent contrapositive:
`~(q V r) -> ~p`

With De Morgan's law:
`(~q ^ ~r) -> ~p`

"If n is not odd and n is not 2, then n is not prime"

## Converse of a conditional statement

The converse of a conditional statement `if p then q`

is obtained by **exchanging
the hypothesis and conclusion**.

Symbollically:
`p -> q`

is `q -> p`

**Note:** A conditional statment and its converse are * not* ligically equivalent.

Example: The converse of the following.

"If it walks like a duck and it talks like a duck, then it is a duck"

`p`

= "it walks like a duck"
`q`

= "it talks like a duck"
`r`

= "it is a duck"

`(p ^ q) -> R`

Converse:
`r -> (p ^ q)`

"If it is a duck, then it walks like a duck and it talks like a duck"

## Inverse of a conditional statement

The inverse of a conditional statement of the form `if p then q`

is obtained
by **negating the hypothesis and conclusion**.

Symbolically:
`p -> q`

is `~p -> ~q`

**Note:** A conditional statement and * its inverse* are

**not**logically equivalent.

**Note:**The converse and the inverse of a conditional statement

**are**logically equivalent.

Example: Write the inverse for the following.

"If n is prime, then n is odd or n is 2"

`p`

= "n is prime"
`q`

= "n is odd"
`r`

= "n is 2"

`p -> (q V r)`

Inverse:
`~p -> ~(q V r)`

`~p -> (~q ^ ~r)`

by De Morgan's law

"if n is not prime, then n is not odd and n is not 2"

## Only if form

If `p`

and `q`

are statement variables, "`p`

only if `q`

" means "if not `q`

then not `p`

",
or equivalently "if `p`

then `q`

".

If `p`

occurs, then `q`

must also occur.

Example: Write the equivalent if-then statement.

"Today is Thanksgiving only if tomorrow is Friday"

`p`

= "Today is Thanksgiving"
`q`

= "Tomorrow is Friday"

Equivalent versions:

- "If today is Thanksgiving, then tomorrow is Friday"
- "If tomorrow is not Friday, then today is not Thanksgiving"

## Biconditional statement

This is a combination of a conditional statement and its converse.

Given statement variables `p`

and `q`

, the biconditional of `p`

and `q`

is
`p if, and only if, q`

or `p iff q`

and is denoted by `p <-> q`

.

- It is true if both
`p`

and`q`

have the same truth values and is false if

`p`

and `q`

have opposite truth values

Truth table:

p | q | p <-> q |
---|---|---|

T | T | T |

T | F | F |

F | T | F |

F | F | T |

Knowing this, then `p <-> q`

has the same truth values as `(p -> q) ^ (q -> p)`

.
This is the converse of the conditional statement.

**Note:** This equivalence is very important for converting complex forms with biconditionals
to convert them into simpler forms.

Example: Write the equivalent statement as a conjunction of two if-then statements.

"This integer is even if, and only if, it equals twice some integer"

`p`

= "This integer is even"
`q`

= "This integer equals twice some integer"

"If this integer is even, then it equals twice some integer and if this integer equals twice some integer, then it is even"

- In order of operations
`<->`

is co-equal with`->`

- As with
`^ V`

the only way to indicat eprecedence is with parentheses

- As with