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### 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

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Logic
High School Logic

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