Sure, let's rewrite each given statement using the appropriate mathematical symbols:
### Part a)
"There exists a positive number [tex]\( x \)[/tex] belonging to the set [tex]\( \mathbb{R} \)[/tex] such that [tex]\( x^2 = 5 \)[/tex]."
This can be expressed as:
[tex]\[ \exists x \in \mathbb{R}, \, x > 0 \, \wedge \, x^2 = 5. \][/tex]
### Part b)
"For every positive number [tex]\( M \)[/tex] there is a positive number [tex]\( N \)[/tex] such that [tex]\( N < \frac{1}{M} \)[/tex]."
This can be expressed as:
[tex]\[ \forall M > 0 \, \exists N > 0, \, N < \frac{1}{M}. \][/tex]
### Part c)
"There exists [tex]\( m \)[/tex] which belongs to the set [tex]\( M \)[/tex]."
This can be expressed as:
[tex]\[ \exists m \in M. \][/tex]
