While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter.
So how do I know if something is a mathematical statement or not?
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$\begingroup$You will know that these are mathematical statements when you can assign a truth value to them. That is, if you can look at it and say "that is true!" or "that is false!" then it is a mathematical statement.
It doesn't mean anything else, it doesn't require numbers or symbols are anything commonly designated as "mathematical."
$\endgroup$ 1 $\begingroup$A sentence is called mathematically acceptable statement if it is either true or false but not both.
1. All primes are odd numbers.
2. Two plus two is four.
In the above sentences 1. is false and 2. is true and hence both of them are mathematical statements.
3. The sum of $x$ and $y$ is greater than 0.
4. Tomorrow is Friday.
Above sentences 3. and 4., for both of them we cannot say whether they are true or false. In 3. unless we know the value of $x$ and $y$ we cannot say anything about whether the sentence is true or false. Same goes for 4..
5. There are 40 days in a month.
We have not specified the month in the above sentence but then too we know that since there is no month which have more than 31 days so the sentence is always false regardless what month we are taking. Hence it is a statement.
In your problem
Statement a. and c. are not mathematical statements because it may be true for one case and false for other. (See my given sentences 3. and 4.)
Statement b. and d. are not mathematical statements because they are just expressions. Here too you cannot decide whether they are true or not.
Statement e. is a mathematical statement because it is always true regardless what value of $t$ you take. (See my given sentences 5.)
A statement (or proposition) is a sentence that is either true or false. In your examples, which ones are true or false and which ones do not have such binary characteristics, i.e they cannot be described as being true or false? For example, me stating every integer is either even or odd is a statement that is either true or false. But $5+n$ is just an expression, is it true or false? We can't assign such characteristics to it and as such is not a mathematical statement.
$\endgroup$ 4 $\begingroup$If there is no verb then it's not a sentence. It can't be true or false.
"Giraffes that are green"
"Giraffes that are green" is not a sentence, but a noun phrase. It cannot be true or false.
"Giraffes that are green are more expensive than elephants." is a complete sentence. It can be true or false.
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