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Abstract Algebra

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Quantifers in Mathematical Proofs

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In mathematical proofs, the word quantifiers usually refers to expressions like for all and there exists. There are standard notations for such expressions, coming from logic.

For example $\forall$ stands for for all, $\exists$ stands for there exists, $\exists!$ stands for there exists a unique, $\nexists$ stands for there exits no.

Many mathematicians prefer using the words instead of the symbols when writing proofs, however a knowledge of the formal logical properties of quantifiers can be very useful, or even necessary, when making a rigorous argument.

Let ${\mathbb Z}$ be the set of all integers and ${\mathbb R}$ the set of all real numbers.

Consider the following statements.

Which one is TRUE?

A

a function $f:{\mathbb R}\rightarrow {\mathbb R}$ is continuous at every $x\in{\mathbb R}$ if and only if $\forall x\in{\mathbb R}$, $\forall \varepsilon >0$, $\exists \delta>0$, such that $\forall h\in{\mathbb R}$ with $|h|<\delta$, we have $|f(x)-f(x + h)|<\varepsilon$.

B

a function $f:{\mathbb R}\rightarrow {\mathbb R}$ is continuous at every $x\in{\mathbb R}$ if and only if $\forall \varepsilon >0$, $\exists \delta>0$ such that $\forall x\in{\mathbb R}$, $\forall h\in{\mathbb R}$ with $|h|<\delta$, we have $|f(x)-f(x + h)|<\varepsilon$.

C

a function $f:{\mathbb R}\rightarrow {\mathbb R}$ is uniformly continuous if and only if $\forall \varepsilon >0$, $\forall x\in{\mathbb R}$, $\exists \delta>0$ such that $\forall h\in{\mathbb R}$ with $|h|<\delta$, we have $|f(x)-f(x + h)|<\varepsilon$.

D

$\forall n\in{\mathbb Z}, \exists!d\ge0$ such that $2^d$ divides $n$

E

$\nexists n\in{\mathbb Z}$ such that $\nexists m\ge 2$ with $m$ dividing $n$