T or F statements

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March 29th 2009, 10:04 AM
relyt

T or F statements

Find if statements are T or F and give explanation?

Am I on the right tracK

$(\forall x \in R) (x >=$ -5)
False. Not all real numbers are greater than or equal to -5. An example would be x=-4

$(\exists z \in R) (z < 1)$
True. There is, at least one, real number z that is less than one. An example: z = -1

$(\exists m \in R) (\forall x \in R) (xm = m)$
False. For example, let m=8 and x = -1. -8 != 8.

$(\exists t \in R) (\forall x \in R) (t < x )$
False. Let x = 1 and t=10
March 29th 2009, 10:41 AM
Plato

Quote:

Originally Posted by relyt

Find if statements are T or F and give explanation?
$(\forall x \in R) (x >= –5)$
False. Not all real numbers are greater than or equal to -5. An example would be x=-4
Where did the $\color{red}-5$ come from?

$(\exists t \in R) (\forall x \in R) (t < x )$
False. Let x = 1 and t=10

You need a more general counter-example.
Note that $\color{red}(\forall t)[(t-1)<t]$
March 29th 2009, 11:00 AM
relyt

Sorry. That should be -5 and not 5 for the first statement
March 29th 2009, 11:48 AM
HallsofIvy

Quote:

Originally Posted by relyt

Find if statements are T or F and give explanation?

Am I on the right tracK

$(\forall x \in R) (x >=$ -5)
False. Not all real numbers are greater than or equal to -5. An example would be x=-4

$(\exists z \in R) (z < 1)$
True. There is, at least one, real number z that is less than one. An example: z = -1

$(\exists m \in R) (\forall x \in R) (xm = m)$
False. For example, let m=8 and x = -1. -8 != 8.

This says "there exist m". what if m= 0?

Quote:

$(\exists t \in R) (\forall x \in R) (t < x )$
False. Let x = 1 and t=10

As in the previous one (which is true) you can't just pick t to get a counter example. For all t, what happens if you take x= t?
March 29th 2009, 12:27 PM
relyt

Ok, thanks.

So the first one is false and the rest are true.

For the "there exists" ones....as long as there is one number that makes the statement correct, then it is true?
March 29th 2009, 12:43 PM
Plato

Quote:

Originally Posted by relyt

Ok, thanks.
So the first one is false and the rest are true.

For the "there exists" ones....as long as there is one number that makes the statement correct, then it is true?

Absolutely NOT!
The first is FALSE. $x=-6<-5$
Second is TRUE. $z=0<1$
Third is TRUE. $m=0$ works.
The fourth is FALSE. $(\forall t)[t-1<t]$ there is no smallest number.
March 29th 2009, 03:05 PM
relyt

I think I'm a little confused now. Can we try one more and maybe someone can explain why this is true or false....and maybe what exactly I should look for first. Thanks and I appreciate the responses so far.

$(\forall x \in R) (\exists y \in R) (xy = 0.537)$

So this reads...for all real numbers x, there exists a real number y such that xy = 0.537
March 29th 2009, 03:20 PM
Plato

Quote:

Originally Posted by relyt

I think I'm a little confused now. Can we try one more and maybe someone can explain why this is true or false....and maybe what exactly I should look for first. Thanks and I appreciate the responses so far.
$(\forall x \in R) (\exists y \in R) (xy = 0.537)$
So this reads...for all real numbers x, there exists a real number y such that xy = 0.537

Zero is a real number. So what if $x=0$ ?
What y would make the statement happen?
March 29th 2009, 03:34 PM
relyt

There is no y that would...so it is false
March 29th 2009, 03:36 PM
Plato

Quote:

Originally Posted by relyt

There is no y that would...so it is false

Correct.