$A\subset\mathbb R$ then use quantifier to write statements.

Question

$A\subset\mathbb R$ then use quantifier to write this statements.

1)$A$ has maximum.

2)$A$ does not have maximum.

3)$A$ is bounded.

My try: $3)$$\forall x\in A,\exists M \,||x|\leq M$ $(1)$$\forall x\in A,\exists x_m\in A|x\leq x_m$

Answer 1

1. $\exists x \in A:\forall y\in A \implies x\ge y$
2. $\not\exists x \in A:\forall y\in A \implies x\ge y$
3. $\exists M:A\subset(-M,M)$

Number 2 can be written as: If $\exists x \in \Bbb R$ such that $\forall y \in A$, $ x\ge y$ then $x\not\in A$

Answer 2

Your tries are not correct !

$(1):\exists x_m\in A, \forall x\in A:x\leq x_m$

$(3): \exists M \, \forall x\in A:|x|\leq M$

Answer 3

In both cases (question 1 and 3) you should reverse the order of $\forall$ and $\exists$.

The condition $(\forall x\in A,\exists M\ge0;\,\vert x\vert\le M)$ holds for every subset $A\subset \mathbb{R}$.

By constrast, the condition $(\exists M\ge0;\,\forall x\in A,\,\vert x\vert\le M)$ is the correct formal version of 3).

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To obtain the formal version of statement 2, let's start with "A does have a maximum", that is assertion 1 :

$$\exists M\in A;\,\forall x\in A,\,x\le M$$

It remains to consider the negation of the previous one :

$$\forall M\in A,\exists x\in A;\,x>M$$