I'm trying to solve this mathematical logic problem, can someone please at least give me a tip on how to approach this problem?
The square of any positive real number is a positive real number.
Write down the,
- statement in symbolic form
- the converse of the statement in symbolic form
- the negation of the statement in symbolic form and sentence form
Thank you.
Hint 1: Try thinking of the given sentence as:
For any $x$, if $x$ is a positive real number, then $x^2$ is a positive real number.
Hint 2: Suppose that we have a statement of the form: $$ \text{If $p$ is true, then $q$ is true.}\tag{1} $$ Then the converse of $(1)$ is of the form: $$ \text{If $q$ is true, then $p$ is true.} $$ Furthermore, the negation of $(1)$ is of the form: $$ \text{$p$ is true but $q$ is false.} $$ Hint 3: Recall that negations of universally quantified predicates turn into existentially quantified (negated) predicates. More precisely: $$ \neg[\forall x,~ P(x)] \qquad\text{is equivalent to}\qquad \exists x ~~\neg P(x) $$
Hint: If you define $P(x)$ (or whatever form you use) to mean "$x$ is a positive real number", the first becomes $P(x) \implies P(x^2)$ How do you write the converse of an implication? To write the negation, you can just precede it by $\lnot$, but you may be expected to turn it into a conjuction. How can an implication be false?
First translate the sentence to 'first order English' language:
- For all positive real number, its square is positive.
- For all real number $x$, if $x$ is positive, then $x^2$ is positive.
