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Conditional and Converse Statements

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mister kitten
 
Reply Wed 23 Sep, 2009 04:01 pm
In my Plane Geometry class today we were learning about Conditional and Converse statements.

This is what tripped me up:
------------------------------------------------------------
If x is greater than or equal to 9, then x is greater than 8 (the conditional)

If x is greater than 8, then x is greater than or equal to 9 (its converse)
------------------------------------------------------------
I got confused. At first our teacher only showed us the conditional statement. I thought of 8.5, at first which at the time I thought was the counterexample to the conditional. But then I learned that you have to start with the first part (the 'if')and then move to the second part (the 'then'). My idea applied to the converse statement though, yay learning!

So I thought that this could be applied to other things. But I have very little time to expond on my thoughts (I'm actually supposed to be writing an essay for philosophy class). What do y'all think?
 
Emil
 
Reply Thu 24 Sep, 2009 03:01 pm
@mister kitten,
1. proposition. For all x, if x is greater than or equal to 9, then x is greater than 8.

2. proposition. For all x, if x is greater than 8, then x is greater than or equal to 9.

These are converses of each other. Both are conditionals also called implications.

Formalization

1. prop. (∀x)(x≥9→x>8)

2. prop. (∀x)(x>8→x≥9)

Truth value

1. prop. is true and necessarily so.

2. prop is false and necessarily so. A counter example would be 8.5.

Note that if you change (2) slightly, then it becomes true:

2'. (∀n)(n>8→n≥9) (n is the set of natural numbers.)

What more do you want to know?
 
mister kitten
 
Reply Fri 25 Sep, 2009 12:18 pm
@mister kitten,
I understand that. Thank you. What does the upside down A mean?
and I'm guessing the arrow means that it 'implies' right?
 
kennethamy
 
Reply Fri 25 Sep, 2009 08:05 pm
@mister kitten,
mister kitten;93121 wrote:
In my Plane Geometry class today we were learning about Conditional and Converse statements.

This is what tripped me up:
------------------------------------------------------------
If x is greater than or equal to 9, then x is greater than 8 (the conditional)

If x is greater than 8, then x is greater than or equal to 9 (its converse)
------------------------------------------------------------
I got confused. At first our teacher only showed us the conditional statement. I thought of 8.5, at first which at the time I thought was the counterexample to the conditional. But then I learned that you have to start with the first part (the 'if')and then move to the second part (the 'then'). My idea applied to the converse statement though, yay learning!

So I thought that this could be applied to other things. But I have very little time to expond on my thoughts (I'm actually supposed to be writing an essay for philosophy class). What do y'all think?


All converses are also conditionals (although, not conversely!).

The converse of If A then B, is, If B then A.

Some converses are equivalent: e.g. If a number is divisible by 2, then that number is an even number; and; if a number is an even number, then that number is divisible by 2. Equivalent means that if one statement is true, then the other statement must also be true.

But some converses are not equivalent. e.g. If something is an apple it is a fruit. and; if something is fruit, it is not an apple. Obviously the first is true, but the second is false. So they are not equivalent.
 
Emil
 
Reply Sat 26 Sep, 2009 03:19 pm
@mister kitten,
mister kitten;93582 wrote:
I understand that. Thank you. What does the upside down A mean?
and I'm guessing the arrow means that it 'implies' right?


No, it means "for all". It is predicate logic quantifier. The reversed E in (∃x) means "there exists at least one".

More on symbols here.

---------- Post added 09-26-2009 at 11:21 PM ----------

kennethamy;93667 wrote:
All converses are also conditionals (although, not conversely!).

The converse of If A then B, is, If B then A.

Some converses are equivalent: e.g. If a number is divisible by 2, then that number is an even number; and; if a number is an even number, then that number is divisible by 2. Equivalent means that if one statement is true, then the other statement must also be true.

But some converses are not equivalent. e.g. If something is an apple it is a fruit. and; if something is fruit, it is not an apple. Obviously the first is true, but the second is false. So they are not equivalent.


Right about everything except "All converses are also conditionals (although, not conversely!).". Can you name a converse that is not a conditional? I can't.
 
kennethamy
 
Reply Sat 26 Sep, 2009 03:31 pm
@Emil,
Emil;93783 wrote:
No, it means "for all". It is predicate logic quantifier. The reversed E in (∃x) means "there exists at least one".

More on symbols here.

---------- Post added 09-26-2009 at 11:21 PM ----------



Right about everything except "All converses are also conditionals (although, not conversely!).". Can you name a converse that is not a conditional? I can't.


No. And you are right. All conditionals are the converse of some conditional that is its converse. Wrote too quickly. Thought too slowly. Silly thing for me to say.
 
Emil
 
Reply Sat 26 Sep, 2009 07:42 pm
@kennethamy,
kennethamy;93788 wrote:
No. And you are right. All conditionals are the converse of some conditional that is it's converse. Wrote too quickly. Thought too slowly. Silly thing for me to say.


That's what I thought. Well, not the silly part, but the rest. :p
 
 

 
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