Symbolic Notations of statements.

Question

The problem asks to put the statement into compact form.

The statement is : For every $2\times2$ matrix $A$ there exists a $2\times 2$ matrix $B$ such that $AB \neq BA$.

My original thought was to form the compact form like this : $(\forall A_{2x2})[(\exists B_{2x2}) \mid (AB \neq BA)]$.

But now I am wondering if I should be using a implication instead of the $ \mid $ modifier. Or if it should be written as $(\forall A_{2x2})(\exists B_{2x2})[AB \neq BA]$.

How does one properly format this statement and if it's no trouble could you explain why? Thank you for any help.

Answer 1

You may want to rewrite the sentence as, "For every $A$, if it is a $2\times 2$ matrix, then there exists a $B$ that is a $2\times 2$ matrix and satisfies $AB \neq BA$." Then you are much closer to the solution.

A problem common to all your attempts is that $A_{2\times 2}$ is, from the syntax viewpoint, just the name of a (bound) variable, and bears no relation to $A$, which is another (free) variable. Said otherwise, $\forall A_{2\times 2}$ does not read, "for every $2\times 2$ matrix."

The precise translation into logic depends on what is the universe of discourse. If it is understood that all elements are $2\times 2$ matrices, you don't need to mention it in the formula. Otherwise you have to.