$A(x)$:$x$ is an apple.
$R(x)$: $x$ is red.
$G(x)$: $x$ is green
I need to translate the following sentence into their equivalent logical form using the above dictionary:
$1$. There is at least one red apple.
$2$. Some apples are red.
$3$. At least one apple is red.
The sentence I came up with for all of these is $∃ x,A(x) ∧ R(x)$
Are all these sentences seem different and does my sentence only fit to one of them?
On
Logically, all of the three statements are same and thus have a unique quantified statement.
Each of them can be rewritten as follows:
There exists some $x$ s.t. $x$ is an apple and $x$ is red.
Note: The existential quantifier $\exists $ is used for the phrases like 'for some', 'there exists' and ''there is a' etc.
In quantified notation; it is $\exists x, A(x)\land R(x)$ as obtained by you.
They are all the same and your answer is fine. One might argue that in English the second statement guarantees that there are at least two different red apples, but that is not the normal mathematical way to read the sentence. If you do read it as guaranteeing two different red apples you need a different transcription.