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Math Logic - Determining Truth


Date: 04/13/99 at 19:29:50
From: Ricky De Hamer
Subject: Math Logic

A number divisible by 2 is divisible by 4. I'm suppose to figure out 
the hypothesis, the conclusion, and a converse statement, say whether 
the converse statement is true or false, and if it is false give a 
counterexample. I don't understand.


Date: 04/14/99 at 09:20:07
From: Doctor Peterson
Subject: Re: Math Logic

Hi, Ricky.

You're asking about the terminology of logic, which is important in 
math to help us talk about proofs and how we know something is true. 
Words such as "converse" allow us to talk about our reasoning and see 
whether we are really making sense.

A statement such as "any number divisible by 2 is divisible by 4" 
(I've changed "a" to "any" to clarify the statement a little) can be 
rewritten as

  IF a number N is divisible by 2, THEN the number N is divisible by 4

The hypothesis, or premise, is what is given or supposed, the "if":

  a number N is divisible by 2

The conclusion is what is concluded from that, the "then":

  the number N is divisible by 4

The converse of the statement "IF a THEN b" is "IF b THEN a", turning 
the statement around so that the conclusion becomes the hypothesis and 
the hypothesis becomes the conclusion. In this case, the converse is

  IF a number N is divisible by 4, THEN the number N is divisible by 2

Now we have to consider whether either statement is true. A statement 
and its converse may be either both true, or both false, or one true 
and the other false; knowing whether one is true says nothing about 
whether the other is true. In this case, the original statement is 
false. (This makes me wonder if you copied the problem wrong; it 
doesn't sound like this possibility was considered in the question.) 
How do I know it's false? Because I can give a counterexample: a 
number N for which the hypothesis is true but the conclusion is false. 
Can you see what I can use for N, which is even but not divisible by 
4?

However, the converse is true. See if you can see why. You might just 
try listing lots of numbers that are divisible by 4, and see whether 
they are all even. If all your examples are even, you haven't proven 
anything; but the list may suggest to you a reason why you will never 
be able to find a counterexample. That reason would be the basis of a 
proof.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Discrete Mathematics
High School Logic

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