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
pandqhave 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
^ Vthe only way to indicat eprecedence is with parentheses
- As with