What are the quantifiers in this statement?

Question

Statement: Suppose $V$ is finite-dimensional with $\dim V \ge 2$. Prove that there exist $S,T \in L(V, V)$ such that $ST \ne TS$

I am confused about the first part "Suppose $V$ is finite-dimensional with $\dim V \ge 2$". Would the quantifier be $\exists V(\dim V \ge 2 \to \exists S,T\in L(V, V)(ST\ne TS))$.

Or would the quantifier be $\forall V(\dim V \ge 2 \to \exists S,T\in L(V, V)(ST\ne TS))$.

My first guess was that it would be $\forall$ as the dimension of the vector space can be anything $\ge 2$ and we have to prove for when the dimension is some number $n\ge 2$. But when I say it out like "some number $n\ge 2$", I feel like the quantifier would be $\exists$.

Could someone explain which one would it be and why?

In mathematics, $[A(x){\implies}B(x)]$ implicitly means $\forall x,[A(x){\implies}B(x)].$ On the other hand, $[A(x){\kern.6em\not\kern-.6em\implies} B(x)]$ implicitly means $\exists x,[A(x){\kern.6em\not\kern-.6em\implies} B(x)].$ — ryang, Aug 04 '22 at 10:54

score 1 · Accepted Answer · answered Aug 04 '22 at 10:00

1

If nothing is said about a variable, then the implicit quantifier is always "for all".

For example, how would you interpret the following sentence:

Suppose $x$ is a positive real number. Then there exists some value $y$ such that $0<y<x$.

Would you say that the above sentence only guarantees the existence of one such $x$, or that it is true for all such $x$?

answered Aug 04 '22 at 10:00

5xum

123,496
6
128
204

I would interpret that as for all. Saying that the statement is true for all values of $x$. I see your point. Because $x$ can be any value in the set of positive real numbers. – Seeker Aug 04 '22 at 10:02

What are the quantifiers in this statement?

1 Answers1

Linked