Here are some "concepts" that are confusing me:
The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case $x$ is free while in the second case $x$ is a bound variable. Now for these two assertions:
If $x>2$ then $x>3$.($x$ is understood to be a real number).
For every $n$ if $n>2$ then $n>3$.
Is the $x$ in the first assertion free while the $n$ in the second assertion bound ? The Handbook of Mathematical Discourse states that in the first assertion $x$ is actually universally quantified like the second assertion. Can someone elaborate on that ?
Also, when proving them, their proofs are exactly the same except in the second one we add "let n be arbitrary"(How to Prove it). So do the two assertions differ in their Logical structure ?
Now for definitions:
A number is even if it is divisible by $2$.
The number is even if it is divisible by $2$.
Is the usage of the and a different in mathematical definitions(sorry for the poor example above, if you have any better example that would be appreciated). Also aren't mathematical definitions assertions ?
A number $n$ is even if it is divisible by $2$.
Every number $n$ is even if it is divisible by $2$.
I know that a definition is NOT an assertion and thus cannot be true or false(Right ?). The definition just describes a property(ies) of some mathematical object.
Now for the above two definitons is $n$ a free variable in the first but bound variable in the second ? Also the second definition can be expressed as $\forall n (n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$. Doesn't that mean that a definition in a sense is true or false since it can be expressed by a logical symbols ?
Suppose $n$ is an integer. Then $n$ is even if it is divisible by $2$.
Where is the location of Suppose $n$ is an integer(a precondition) in the logical structure of a definition(after or before the biarrow) ? $\forall n (n\ is\ an\ integer \rightarrow n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$ or $\forall n (n\ is\ an\ integer \land n\ is\ even\ \leftrightarrow n\ is\ divisible\ by\ 2)$ or $\forall n ( n\ is\ even\ \leftrightarrow n\ is\ an\ integer \land n\ is\ divisible\ by\ 2)$
Also when trying to prove "$x$ is even" what exactly should I do ?
Suppose $R$ is a partial order on a set $A$, $B ⊆ A$, and$ b ∈ B$. Then$ b$ is called an $R$-smallest element of B (or just a smallest element if $R$ is clear from the context) if $∀x ∈ B(bRx)$.
How do I express this definiton in logical symbols(are $A$,$b$,$B$, $R$ free or bound?)?What should I do if I want to prove that $z$ is a $H$ smallest element of $M$ ?
Suppose $f : A → B$ and $C ⊆ A$. The set $f ∩ (C × B)$, which is a relation from $C$ to $B$, is called the restriction of f to C, and is sometimes denoted $f|C$. In other words, $f|C = f∩(C × B)$.
What is the location of $f|C$ in the logical structure ?
Also, it would be great to give a list of books that could clarify misconceptions of these type.