Logic: "the cubic root of a rational number is also a rational number"

Question

I was attempting an online logical and mathematical statements self-test from the University of Toronto and came across the following statement in question 1:

The cubic root of a rational number is also a rational number.

We are asked to select an equivalent statement from the provided list. The answer turned out to be:

If $x$ is a rational number, then $\sqrt[3]{x}$ is a rational number.

Why is that?

The given statement seems to be of the form "A is also a B".

A statement fitting that form would be "a feline is also a mammal". But this wouldn't be equivalent to "if $x$ is a mammal, then $x$ is a feline".

I actually translated the given statement as follows, but it wasn't one of the available options:

$\forall x \in \mathbb{Q}, x = \sqrt[3]{y} \Longrightarrow x \in \mathbb{Q}$

Can someone help me identify the flaws in my reasoning, and why the given answer is the correct one?

Answer 1

I think your confusion simply comes from the ambiguity of natural language.

When we say ‘the cubic root of a rational number is also a rational number’, this means ‘for all rational numbers, the cubic root of this number is also a rational number.’ or $\forall x \in \mathbb{Q}, \sqrt[3]{x} \in \mathbb{Q}$. As an implication, we can write this as $x \in \mathbb{Q} \implies \sqrt[3]{x} \in \mathbb{Q}$.

In general, statements of the form ‘$A(B)$ is also a $B$’ can be written as:

$x$ is a $B$ $\implies$ $A(x)$ is a $B$.

(in the above case we have $A$ being the cubic root function and $B$ being ‘a rational number’)

A simpler type of statement is one of the form ‘an $A$ is also a $B$’. This can be written as:

$x$ is a $A$ $\implies$ $x$ is an $B$.

Your statement ‘A feline is also a mammal’ is of this form and so it is clear, as you say, that it is not equivalent to ‘if x is a mammal, then x is feline’.

These two types of statements are fundamentally different. The first type of statement is special because B is referenced on both sides of the implication.

Answer 2

Certainly statements of the form "A is also a B" do not allow us to say "If $x$ is B then $x$ is A" as your feline example shows. However, the original statement is special because the "B" statement occurs twice (once within the statement of A itself). The equivalent statement is not formed by swapping the ordering of A and B as your feline example suggests. It is formed by using the structure of A.

1) The original statement is this (notice that "rational number" appears twice): $$ \underbrace{\mbox{The cubic root of a rational number}}_A \mbox{ is also a }\underbrace{\mbox{rational number}}_B$$

Equivalently:

• "If $y$ is the cubic root of a rational number then $y$ is B."

• "$\underbrace{\mbox{If $x$ is a rational number then its cubic root}}_{\mbox{(this is still refering to A)}}$ is B."

Your mistake is to think the part in the above underbrace is refering to B (i.e., "hey they are just swapping the ordering of A and B") which it is not. The above three statements would be equivalent regardless of B. It is only by accident that B is something that is also mentioned in A. That does not give us freedom to swap orderings of B and A whenever we want.

2) Similarly these are equivalent statements:

• The child of a feline is also a feline.

• If $y$ is the child of a feline then $y$ is a feline.

• If $x$ is a feline then its child is a feline.

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Having said that, that particular question on the test is a bad one because it is false (3 is rational but $3^{1/3}$ is not). That is what motivated my initial comment that said your statement is equivalent to $0=1$.

Answer 3

Let's firstly put aside the completely irrelevant question herein of whether the statements are true; obviously, it's not true in general that a rational number's cube root is rational. Let's look into how we rewrite statements with algebra.

While "a feline is also a mammal" becomes "if $x$ is a feline, then $x$ is a mammal", "the cubic root of a rational number is also rational" means "if $y$ is the cube root of a rational $x$, then $y$ is a rational number". Since this time two objects are named in algebra, it's more natural to rearrange the second example as "if $x$ is a rational number and $y=\sqrt[3]{x}$, then $y$ is a rational number". And it's even more natural to only have one named variable, viz. "if $x$ is a rational number, then $\sqrt[3]{x}$ is a rational number".