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#1 2007-09-13 17:59:07

Identity
Member
Registered: 2007-04-18
Posts: 934

Truth Table

X    Y      X⇒Y
F     T        T
F     F        T
T     T        T
T     F        F

I don't understand why F ⇒ T. A false statement can't imply a true statement, can it???

I imagine that this table should look like the table for &:

X    Y      X&Y
F     T        F
F     F        F
T     T        T
T     F        F

Ugh... Math logic is so dizzy

Last edited by Identity (2007-09-13 22:26:23)

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#2 2007-09-13 19:50:05

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Truth Table

In the X&Y truth table, F&F should be F, not true.

In X⇒Y, if X is false, the whole implication is true, regardless of Y. The implication is said to be vacuously true.

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#3 2007-09-13 21:22:18

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Truth Table

This confused me when I first came across it. If X is false then surely the statement X⇒Y is indeterminate? We don't know what happens to Y when X is true, so we can't say anything.

I got around it by redefining T in my head as 'not false'.

Another way you can do it is to get rid of all ⇒ signs. It's an identity that X⇒Y is the same as ¬XvY, and it's less confusing to state whether that second one is true or false.

Incidentally, that second truth table you have is for iff.


Why did the vector cross the road?
It wanted to be normal.

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#4 2007-09-13 22:08:41

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Truth Table

Thanks, I kind of see where you're going... (and yeah I made a mistake in the second table)

mathsyperson, I don't understand how

. The LHS of the equivalence can be expressed on a venn diagram, but the right... I don't think it can? The left doesn't have a 'then' in it

Last edited by Identity (2007-09-13 22:25:47)

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#5 2007-09-14 00:01:24

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Truth Table

When I said that, I really meant 'in the eyes of a truth table'. They might be interpreted differently, but they get you to the same result.


Why did the vector cross the road?
It wanted to be normal.

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#6 2007-09-14 04:12:37

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Truth Table

So to get this, you really need to see what an if statement says.  An if statement only makes a conclusion when the preconditions are met.

If it is raining, I will get wet when I go outside.

This statement only makes a conclusion when it is raining.  It makes no conclusion at all when it is not raining.  How can you be wrong (false) if you say absolutely nothing?  Simple: you can't be wrong.  The if statement is not wrong when it's preconditions are not met.

Simple examples in math can demonstrate this quite well.

If p is a prime greater than 2, then p is odd.

I would sure hope that is a valid statement.  p is 4 and p is not odd.  What we have here is that F -> F.  Certainly we would not say our if-then statement does not hold because of this example.  Similarly, p is 9 and p is odd.  What we have is that F -> T.  And of course, this does not contradict the statement either.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2007-09-14 04:19:54

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Truth Table

Try this puzzle I just made up myself. wink

Three people make the following the statements:

A: “None of us is lying.”
B: “Every one of us is lying.”
C: “If my name’s not Charlie, then I’m a monkey’s uncle!”

Given that C is not a monkey’s uncle, prove that C’s name is indeed Charlie. tongue

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#8 2007-09-14 04:46:38

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Truth Table

That's just mean Jane... Nicely done.

Edited to add:

labyrinth_puzzle.png


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#9 2007-09-14 05:15:01

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Truth Table

JaneFairfax wrote:

Try this puzzle I just made up myself. wink

Three people make the following the statements:

A: “None of us is lying.”
B: “Every one of us is lying.”
C: “If my name’s not Charlie, then I’m a monkey’s uncle!”

Given that C is not a monkey’s uncle, prove that C’s name is indeed Charlie. tongue

Lol.. uh is this supposed to be a real problem? A and B are cool... but I don't think they help.
C by itself is of the form

, so even if C is not a monkey's uncle, you can't prove that C's name is Charlie.

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#10 2007-09-14 05:26:41

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Truth Table

The puzzle does work, but you need to use the contrapositive of statement C.

For a statement X⇒Y, the contrapositive is that ¬Y⇒¬X.
It's fairly easy to see how this works.

Statements A and B are a roundabout way of establishing that statement C is true.


Why did the vector cross the road?
It wanted to be normal.

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#11 2007-09-14 05:33:30

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Truth Table

Thanks Mathsy. smile

For those who are interested,

.

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#12 2007-09-16 12:42:01

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

Re: Truth Table

I have a web page on logic for electronics people.
First you must learn Karnaugh maps before you will get
the 4 by 4 grids probably, but anyway take a peek at
all the equations for the grids at:
http://johnericfranklin.250free.com/


igloo myrtilles fourmis

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