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Rewrite each statement using [tex]$\exists, \forall, \in$[/tex], and [tex]$\mid$[/tex] as appropriate.

a) There exists a positive number [tex]$x$[/tex] belonging to the set [tex]$\mathbb{R}$[/tex] such that [tex]$x^2 = 5$[/tex].

b) For every positive number [tex]$M$[/tex] there is a positive number [tex]$N$[/tex] such that [tex]$N \ \textless \ \frac{1}{M}$[/tex].

c) There exists [tex]$m$[/tex] which belongs to the set [tex]$M$[/tex].

Sagot :

GinnyAnsweridn
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]
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