Please help with Logical Statements Thinking | Conditional Statement Example

[CokeCan]

Hello everybody. I'm having problems figuring out how to find out the truth value of logical statements.
I hope you can see how I think when I solve these kind of exercises and I hope you can correct me and give me some tips.

Here's an example:

"Is this statement true or false? Explain why."

x²+5 > 3*(1-x) → x²+1 > 10 with x∈R

Conditional/Implication
"p implies q" or "If p then q"

p: x²+5 > 3(1-x)
q: x²+1>10

x²+5 <=> x² = -5, gives x complex solutions, then x∉R

3(1-x) same as 3-3x
3-3x = 0 <=> x = 1
Can't compare with x²+5 solutions, let's just say p is false.

x²+1 <=> x² = -1, gives x complex solutions, then x∉R
Can't compare these with 10, let's just say q is false.

F implies F
Solution: The statement is true??

Does this even make sense? I'm confused guys, please help.

Thank you so much!

 

[stapel]

Here's an example:

"Is this statement true or false? Explain why."

x²+5 > 3*(1-x) → x²+1 > 10 with x∈R

Conditional/Implication
"p implies q" or "If p then q"

p: x²+5 > 3(1-x)
q: x²+1>10

x²+5 <=> x² = -5, gives x complex solutions, then x∉R

This is nonsensical. There is no mathematical basis for saying "(this expression) if and only if (that equation)". Also, how does this statement relate to the exercise?

3(1-x) same as 3-3x
3-3x = 0 <=> x = 1
Can't compare with x²+5 solutions, let's just say p is false.

What do you mean by "comparing"? What are you trying to accomplish?

x²+1 <=> x² = -1, gives x complex solutions, then x∉R
Can't compare these with 10, let's just say q is false.

Again, this is nonsensical.

F implies F
Solution: The statement is true??

What did you get when you graphed the two related functions, f(x) = x^2 + 3x + 2 and g(x) = x^2 - 9? Was g(x) positive everywhere that f(x) was positive, so that f(x) > 0 implied g(x) > 0?

Does this even make sense?[/QUOTE]
No.

 

[JeffM]

Hello everybody. I'm having problems figuring out how to find out the truth value of logical statements.
I hope you can see how I think when I solve these kind of exercises and I hope you can correct me and give me some tips.

Here's an example:

"Is this statement true or false? Explain why."

x²+5 > 3*(1-x) → x²+1 > 10 with x∈R

Conditional/Implication
"p implies q" or "If p then q"

p: x²+5 > 3(1-x)
q: x²+1>10

x²+5 <=> x² = -5, gives x complex solutions, then x∉R

3(1-x) same as 3-3x
3-3x = 0 <=> x = 1
Can't compare with x²+5 solutions, let's just say p is false.

x²+1 <=> x² = -1, gives x complex solutions, then x∉R
Can't compare these with 10, let's just say q is false.

F implies F
Solution: The statement is true??

Does this even make sense? I'm confused guys, please help.

Thank you so much!

You certainly are confused.

Let's translate the proposition to be evaluated into less formal terms

If it is true that \(\displaystyle x^2 + 5 > 3(1 - x),\) then it is necessarily true that \(\displaystyle x^2 + 1 > 10.\)

That whole sentence is to be assessed for whether it is necessarily true or not.

In other words, if the proposition is true, you should be able to prove the conclusion that x^2 + 1 > 10 from the premise
that x^2 + 5 > 3(1 - x).

If the proposition is false, you should be able to show a real number that makes x^2 + 5 > 3(1 - x) true and x^2 + 1 > 10 false.

 

[CokeCan]

That whole sentence is to be assessed for whether it is necessarily true or not.

In other words, if the proposition is true, you should be able to prove the conclusion that x^2 + 1 > 10 from the premise
that x^2 + 5 > 3(1 - x).

If the proposition is false, you should be able to show a real number that makes x^2 + 5 > 3(1 - x) true and x^2 + 1 > 10 false.

Woof, let's put 2 bones in there.

2^2 + 5 > 3(1 - 2)
9 > -3
True

2^2 + 1 > 10
5 > 10
False

A: The proposition is false.


Rhhff.. It's so simple. I don't know how my human didn't figure that out.
Thank you for your answers. Woof!!