Truth of a Biconditional Statement

Date: 11/08/2005 at 17:01:08
From: Carl
Subject: Truth of a biconditional statement

Let p represent x = 0, and let q represent x + x = x.  Write the 
biconditional p <-> q in words.  Decide whether the biconditional is true.

I know that the converse of this biconditional is true: "x + x = x if
and only if x = 0".  How do I know if the biconditional is true?

Date: 11/10/2005 at 14:59:24
From: Doctor Achilles
Subject: Re: Truth of a biconditional statement

Hi Carl,

Thanks for writing to Dr. Math.

This is a good question.  You have the correct wording for the
biconditional.  To decide if it's true, two things have to be true.

  1) If x = 0, then x + x = x

  2) If x + x = x, then x = 0

To test the first one, let's let x = 0.  If we do that, then the equation

  x + x = x

becomes

  0 + 0 = 0

which is true.  So we're halfway there.

To test the second part, let's grant that

  x + x = x

and do some algebra.  Specifically, let's subtract x from both sides
of the equation.  What does that give us?

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/