Is "$1$ is not an integer" a contradiction? I learnt that a contradiction is a statement which is always false regardless of the truth values of the individual statements that it is made up of.
Now "$1$ is not an integer" is a single statement which is false. Thus there is no question of it being made up of several statements. Can I say that it is a contradiction?
Thanks.
 A: Technically, you need both the statement "1 is not an integer" and the statement "1 is an integer" to have a logical contradiction.
However, within any 'normal' mathematical context, "1 is an integer" is assumed to be true: all working mathematicians use and know "1" to refer to a specific integer. So, if within that context you ever derive "1 is not an integer" based on some assumption, then you can immediately say that that assumption leads to a contradiction; Having to spell out that "1 is an integer" really seems to be rather pedantic. 
In formal logic proofs, sure, everything needs to be explicit, but in normal practical mathematical contexts, I would say there is really no need to explicitly get "1 is an integer" before you can say you have a contradiction.
A: No, a contradiction is either two statements that contradicts each other or a statement that is a conjunction of two such statements.
If you want it to be a contradiction you must also have the statement "1 is an integer". Then you have two statements that contradicts each other.
