Express using logic symbols:

Question

I have to also decide whether it is true or not after I express it in logic symbols. This is what I have so far..Am I correct? and I don't know how to do a)..

a) There is a smallest positive number

-no idea how to approach this one? Perhaps somehow denote this as infinity? and how can I even state a true/false value? Is this even a statement?

b) Every integer is a product of two integers

True: (x,y belong to the set of integers): ∀x∀y(x * y = (set of integers))

c) The equation x^2+y^2 = 1 has a solution (x,y) in which both x and y are natural numbers

True: (x,y belong to the set of natural numbers): ∀x∀y(x^2+y^2 = 1)

d) Every real number can be written as a difference of two positive real numbers

True: (x,y belong to the set of real numbers): ∀x∀y(x-y = (set of real numbers))

Thank you very much!

Answer 1

There is a smallest positive number $\epsilon > 0$ such that $\forall a, b \in R : a - b > \epsilon$. This is denoted as $\mathbf{epsilon}$.

The second one would be similar to yours:

$\forall x \in \mathbb{Z}[\exists a, b \in \mathbb{Z} : ab = x]$

I think these provide enough hints for you to finish.

Answer 2

I will show the first problem for you. We are talking about a smallest positive number, let's call this number $a$. Then the number $a$ has two properties:

$\bullet$ $a$ is positive, or in formal language: $a \in \mathbb{R}_+$.
$\bullet$ $a$ is the smallest positive number, so any other positive number $b$ is greater or equal to $a$.

We can then write the statement "there exists a smallest positive number" as follows: $\exists a \in \mathbb{R}_+ \forall b \in \mathbb{R}_+ : a \leq b$.

Can you solve the other problems now?

Answer 3

Here are the two most direct ways to say that there is no smallest positive number:

$$\begin{align*} &\neg\exists x\forall y(y>0\to y\ge x)\\ &\forall x\big(x>0\to\exists y(y>0\land y<x)\big) \end{align*}$$

In words:

• There is no number $x$ such that whenever $y$ is positive, $y$ is at least as large as $x$.
• For every positive number $x$ there is a positive number $y$ that is smaller than $x$.

Your formula for (b) does not say what you want it to say: it says that the product of any two integers whatsoever is the set of integers. That’s clearly not right: the product is an integer, not the whole set of integers. But even if you changed it to $\forall x\forall y(xy\in\Bbb Z)$, which is at least a true statement (meaning that the product of any $x$ and $y$ is an integer), it wouldn’t say what you want. You want to say that whenever $x$ is an integer, there are integers $y$ and $z$ whose product is $x$. Notice the existence statement there: you’re almost certainly going to want some existential quantifiers. And in fact the English translates pretty directly into symbols: $\forall x\exists y\exists z(yz=x)$. Note, though, that this works only if it’s understood that that $x,y$, and $z$ range over the integers. If not, you have to specify that as well, either as

$$\forall x\in\Bbb Z\exists y\in\Bbb Z\exists z\in\Bbb Z(yz=x)$$

or, more formally, as

$$\forall x\Big(x\in\Bbb Z\to\exists y\exists z(y\in\Bbb Z\land z\in\Bbb Z\land yz=x\Big)\;.$$

Your answer to (c) is also wrong, I’m afraid: you’ve said that no matter what real numbers you substitute for $x$ and $y$, $x^2+y^2$ will be $1$. The sentence that you’re supposed to be translating makes a much weaker claim: it says that there is at least one pair of numbers $x$ and $y$ such that $x^2+y^2=1$. That’s $\exists x\in\Bbb R\exists y\in\Bbb R\left(x^2+y^2=1\right)$.

In (d) you’ve made the same kind of mistake as in (b). It makes no sense to say that $x-y$ is equal to the whole bloomin’ set of real numbers: it’s just one real number! And when we fix that, so that it says $\forall x\in\Bbb R\forall y\in\Bbb R(x-y\in\Bbb R)$, we’re just saying that if you take the difference of two real numbers, you get a real number. That’s not the same as saying that if you start with any real number $x$, you can find two real numbers whose difference is $x$, which is what you’re supposed to be saying. That requires a couple of existential quantifiers: $$\forall x\in\Bbb R\exists y\in\Bbb R\exists z\in\Bbb R(y-z=x)\;.$$ Structurally this is exactly like (b).

Answer 4

A) "There is a smallest positive number" should be

$$ \exists N \in PositiveNumbers : smallest(N) $$

Any clarification of the concept of numbers and smallest is speculation and thus not logically correct.