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How would I translate the following from english to a symbolic sentence with quantifiers.

The universe of discussion is all real numbers.

Every integer is greater than some integer.

I did the following

$\forall x,\exists y$, (if x is an integer and y is an integers then $x>y$)

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2 Answers 2

up vote 4 down vote accepted

Since the domain of discourse is the reals, your currently suggested translation would be satisfied by taking $y$ a non-integer. And probably it is intended that you use logical symbols as much as possible.

There are many equivalent ways to do the job. Use $\text{Int}(t)$ as an abbreviation for "$t$ is an integer." Then we can use $$\forall x(\text{Int}(x)\implies \exists y(\text{Int}(y)\land x\gt y)).$$

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I see because I made an if then statement my antecedent could be false but my statement x>y could work rendering the logic not correct. –  Fernando Martinez Jun 11 at 18:27
    
Yes, you have explained quite precisely why having "if $y$ is an integer" produces an improper translation. –  André Nicolas Jun 11 at 18:29

You can use something like this:

$\forall x \exists y (x \in \mathbb{Z} \implies (y \in \mathbb{Z}) \land (y < x) ) $

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2  
This doesn't look right. You're saying that for all real $x$, it must be that $x \in \mathbb Z$. –  Théophile Jun 11 at 18:21
    
Sure! I will correct... –  HilarioFernandes Jun 11 at 18:23

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