# Help in Math: What is Statement?

I'm preparing TMUA and reading the material about TMUA provided by authority. But there are several confusing points.
(1)Is 'positive integer x is divisible by 2' a statement?
(2)Is 'if x is a positive integer, x is divisible by 2' a statement?
(3)Is 'if positive integer x is divisible by 3, x is divisible by 2" a statement?
What is statement? A sentence that is either true or false but not both? If so, (1) (2) (3) cannot be statements since there are some values of x that make the sentence true but others make the sentence false.
Original post by ECFDPB
I'm preparing TMUA and reading the material about TMUA provided by authority. But there are several confusing points.
(1)Is 'positive integer x is divisible by 2' a statement?
(2)Is 'if x is a positive integer, x is divisible by 2' a statement?
(3)Is 'if positive integer x is divisible by 3, x is divisible by 2" a statement?
What is statement? A sentence that is either true or false but not both? If so, (1) (2) (3) cannot be statements since there are some values of x that make the sentence true but others make the sentence false.

Do you have the question or context (notes) of where these statements come from?
Original post by mqb2766
Do you have the question or context (notes) of where these statements come from?

I make up (2) (3), but on the note, (1) is considered as 'not a sentence'
Here's the note https://uat-wp.s3.eu-west-2.amazonaws.com/wp-content/uploads/2024/05/16150136/Notes_on_Logic_and_Proof_May2024.pdf
You can see (1) at the end of page 7
Original post by mqb2766
Do you have the question or context (notes) of where these statements come from?
I wrote an email to authority but the authority didn't respond to me
Original post by ECFDPB
I make up (2) (3), but on the note, (1) is considered as 'not a sentence'
Here's the note https://uat-wp.s3.eu-west-2.amazonaws.com/wp-content/uploads/2024/05/16150136/Notes_on_Logic_and_Proof_May2024.pdf
You can see (1) at the end of page 7

For (1) thats pretty much covered in the notes, so preceding sentence and first paragraph of page 8.
Original post by ECFDPB
I wrote an email to authority but the authority didn't respond to me

For the red statement, it would be false as a^2<b^2 does not always imply 0<a<b.
Original post by mqb2766
For (1) thats pretty much covered in the notes, so preceding sentence and first paragraph of page 8.

I know! I mean, according to notes, it seems like that anything in the form of 'if A then B' is a statement. But if so, what's the difference between (1) and (2)? Why (1) is not a statement but (2) is a statement?
Original post by ECFDPB
I know! I mean, according to notes, it seems like that anything in the form of 'if A then B' is a statement. But if so, what's the difference between (1) and (2)? Why (1) is not a statement but (2) is a statement?

On page 24 of the notes they discuss statements of the form IF A THEN B, and (1) is covered in the notes on page 8. Loosely, if all elements of A are also in B, then the statement IF A THEN B is true, otherwise its false. So you just need one counter example to negate it. For (1), as in the notes and as you note above, its an expression.
Original post by mqb2766
On page 24 of the notes they discuss statements of the form IF A THEN B, and (1) is covered in the notes on page 8. Loosely, if all elements of A are also in B, then the statement IF A THEN B is true, otherwise its false. So you just need one counter example to negate it. For (1), as in the notes and as you note above, its an expression.

Can I consider all expression in the form of ' if A then B ' as a statement?🤔 Vert strange I have to say
Original post by ECFDPB
Can I consider all expression in the form of ' if A then B ' as a statement?🤔 Vert strange I have to say

Assuming A and B are suitably defined, then yes. You can check whether all elements of A are in B.

As they say in the notes, (1) is really talking about a single element which is a positive integer so to say whether its divisible by 2 or not, you need to know what value it is. In (2), A is the set of all positive integers and the simple counter example, 1 is in A but not B (divisible by 2), means its false.