Background:
Exercise 5: If $a=bc$ with $a\neq 0$ and $b$ and $c$ nonunits, show that $a$ is not an associate of $b$
Questions:
For the above question, is the contrapositive of the statement of the above exercise: if $a$ is an associate of $b$, and $a=bc$ with $a\neq 0$ then $b$ or $c$ is a unit.
I am haveing trouble with the latter part of the statement which has the following:
$a=bc$
$a\neq 0$
$b$ is a not a unit
$c$ is a not a unit
$b$ is a not a unit, $c$ is a not a unit, I can negate it to become a disjunction. But "$a=bc$ with $a\neq 0$", do I treat the "with" here as an and, or do i simply consider "$a=bc$ with $a\neq 0$" as one single statement?
Thank you in advance
Given:
((= with ≠0) and ( and nonunits)) $\rightarrow$ ( is not an associate of ).
The contrapositive is:
$\neg$ ( is not an associate of ) $\rightarrow$ $\neg$ ((= with ≠0) and ( and nonunits)).
$\neg$ ( is not an associate of ) $\rightarrow \neg$ ((= $\land$ ≠0) $\land$ ( nonunit $\land$ nonunit))
( is an associate of ) $\rightarrow$ ($\neg$(= $\land$ ≠0) $\lor$ $\neg$( nonunit $\land$ nonunit))
( is an associate of ) $\rightarrow$ $\neg$(=) $\lor$ $\neg$(≠0) $\lor$ ( unit $\lor$ unit)
( is an associate of ) $\rightarrow$ ($\neq$) $\lor$ (=0) $\lor$ ( unit $\lor$ unit)
( is an associate of ) $\rightarrow$ ($\neq$) $\lor$ (=0) $\lor$ ( unit) $\lor$ ( unit)
I would write: If is an associate of then $a\neq bc$ or $a=0$ or $b$ is a unit or $c$ is a unit.
I don't know the actual content of your statement so it may make sense to stop applying the negations earlier to have your contrapositive statement that you would like to prove. The statement as you wrote it is missing some if/then structure.