How to use basic logical symbols

Asked Sep 10 '20 at 18:22

Active Sep 10 '20 at 18:28

Viewed 94 times

I have a question about basic logical symbols.

You can only use these symbols and I am trying to use them for this question, where $x \in \mathbb R$:

$⇒$, $=$, $⇔$

$a)$ $1 < x$ symbol $1 ≤ x$

$b)$ $x < 0$ symbol $0 < x^2$

$c)$ $0 < x^2$ symbol $x ≠ 0$

My answers:

$a)$ it can't be $⇔$ because you can't go from $1 ≤ x$ to $1 < x$ and they are not exactly the same so it must be $⇒$

$b)$ is $⇒$ as well because if $x = -1$ or any negative number it will translate to positive in $0 < x^2$ and because in the first one it is required to be $x < 0$ it cant be $0$. $⇔$ cant be used because $0 < x^2$, $x$ can be positive but it has to be negative to apply to $x < 0$

$c)$ here I am not that sure, but I think it should be $⇔$ because it works from both sides

edited Sep 10 '20 at 18:28

asked Sep 10 '20 at 18:22

zellez11

For (c), do you mean "$x^2<0$ symbol $0\neq 0$"? – Shaun Sep 10 '20 at 18:24
or possibly $x \ne 0$ instead of $0 \ne 0$? – player3236 Sep 10 '20 at 18:27
yeah sry clumsy mistake on my part – zellez11 Sep 10 '20 at 18:28
4

Well, good job! You got them all correct. (Also note that $=$ is not a logic symbol.) – player3236 Sep 10 '20 at 18:31
2

The second one could also be $=$, i.e., $x < 0 = 0 < x^2$. But I'm guessing $=$ is a typo? Is it supposed to be $\Leftarrow$? – Théophile Sep 10 '20 at 18:31
you mean in my description of the question I just looked at the question it is this symbols $⇒$, $=$, $⇔$ – zellez11 Sep 10 '20 at 18:34
1

That's curious. If it's not a typo, then it's a valid answer to (b). It's different from the other symbols in that it's only linking the zeroes ($0=0$), whereas the other symbols are linking whole statements, i.e., $(1<x) \implies (1 \le x)$. – Théophile Sep 10 '20 at 18:36
so $=$ is a valid answer to b ? – zellez11 Sep 10 '20 at 18:37
= is not a valid answer to b. Consider the case where x=1. Then x<0 is false, but 0<x^2 is true. – user400188 Sep 11 '20 at 05:01
ok so would you consider my answers correct then ? – zellez11 Sep 11 '20 at 05:03
All of your answers are correct. – user400188 Sep 11 '20 at 05:52

How to use basic logical symbols

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