Does every natural number n being even and prime imply that there is a unique n?

[cragar]

Homework Statement

Express the following English sentence as a symbolic statement.
Do not use the there exist symbol
“There is a unique natural number n which is both even and prime.”

We also have properties like
let P(n) be the property that n is prime, L(n, m) the relation that n < m, and E(n) the property that n is even.

The Attempt at a Solution

\forall n (E(n) and P(n) and n&lt;3)

 

[Jeff Rosenbury]

Perhaps I'm under-thinking this: 2.

It doesn't convey all the information of the original, but any sentence that does will evaluate to 2. And most will reduce to 2 as well.

 

[PeroK]

Homework Statement

Express the following English sentence as a symbolic statement.
Do not use the there exist symbol
“There is a unique natural number n which is both even and prime.”

We also have properties like
let P(n) be the property that n is prime, L(n, m) the relation that n < m, and E(n) the property that n is even.

The Attempt at a Solution

\forall n (E(n) and P(n) and n&lt;3)

What about first changing the English version into saying what is not true?

I interpret your symbolic statement (attempted solution) to mean: all natural numbers are even, prime and less than 3.

 

[haruspex]

"There is ..."

How do you write that in symbols?

 

[cragar]

Would it work to say that n \in \mathbb{N}
n \in \mathbb{N} : (P(n) \&amp; E(n))

 

[haruspex]

Would it work to say that n \in \mathbb{N}
n \in \mathbb{N} : (P(n) \&amp; E(n))

No, you must represent "there is" somehow. Are you not aware of a symbol for that?

 

[cragar]

the only symbol I am aware of is there exists \exists but we were told tonot use it in the problem.
Im not sure of a symbol for that

 

[haruspex]

the only symbol I am aware of is there exists \exists but we were told tonot use it in the problem.
Im not sure of a symbol for that

Sorry, I overlooked that.
In that case, as others have suggested, you can get around it by making explicit reference to that number. One part of your answer will express that 2 has these properties. What will the rest of your answer state?

 

[PeroK]

It seems an obvious approach to me that if you're given a statement "there exists ..." and you're told not to use the $\exists$ symbol, then the first step is to rephrase the statement to avoid the phrase "there exists". That, it seems to me, is logical!

As a start, you could think of a statement like "not all primes are odd". This starts to take you in the right direction, I believe.