Sure, let's rewrite each statement using the appropriate symbolic notation involving existential quantifiers ([tex]$\exists$[/tex]), universal quantifiers ([tex]$\forall$[/tex]), set membership ([tex]$\in$[/tex]), and logical conditions (|).
### a)
"There exists a positive number [tex]\(x\)[/tex] belonging to the set [tex]\(R\)[/tex] such that [tex]\(x^2 = 5\)[/tex]."
Using the appropriate symbolic notation, this can be written as:
[tex]\[
\exists x \in \mathbb{R}, \; x > 0 \; | \; x^2 = 5
\][/tex]
### b)
"For every positive number [tex]\(M\)[/tex] there is a positive number [tex]\(N\)[/tex] such that [tex]\(N < \frac{1}{M}\)[/tex]."
Using the appropriate symbolic notation, this can be written as:
[tex]\[
\forall M \in \mathbb{R}, \; M > 0, \; \exists N \in \mathbb{R}, \; N > 0 \; | \; N < \frac{1}{M}
\][/tex]
### c)
"There exists [tex]\(m\)[/tex] which belongs to the set [tex]\(M\)[/tex]."
Using the appropriate symbolic notation, this can be written as:
[tex]\[
\exists m \in M
\][/tex]
These symbolic statements succinctly capture the logical structure of the given sentences.
