Help me please. ($p\Rightarrow q$)
What is the reciprocal, the negation and the Contrapositive of the following affirmation
$$ \text{A Lannister always pays his debts} $$ the negation is: There was one case where a Lannister did not pay,
reciprocal($q\Rightarrow p$) is: If you pay your debts you are a lannister? this is true?
Contrapositive ($\sim q\Rightarrow \sim p$).If you do not pay your debts you are not a lannister? this is true?
help me please.
The initial statement SEEMS to be a sentence $$\forall X\left (X\mbox{ is a Lannister}\Longrightarrow X\mbox{ pays his debts}\right ) $$ Read this as: For every $X$, if $X$ is a Lannister, then $X$ pays his debts.
To negate the statement, it becomes $$\exists X: X\mbox{ is a Lannister and } X\mbox{ does not pay his debts} $$ Read this as: There exists an $X$ such that $X$ is a Lannister and $X$ doesn't pay his debts.
To form the contrapositive, which is logically Equivalent to the initial statement: $$\forall X\left (X \mbox{ does not pay his debts}\Longrightarrow X\mbox{ is not a Lannister}\right ) $$ Read this as: For every $X$, if $X$ does not pay his debts, then $X$ is not a Lannister.
There's the part "..always pays his debts". $$\forall X,\forall Y\left (X\mbox{ is a Lannister and }Y\mbox{ is a debt of $X$}\Longrightarrow X\mbox{ pays } Y\right ) $$ The negation becomes: $$\exists X,\exists Y : X\mbox{ is a Lannister and } Y\mbox{ is a debt of }X\mbox{ and } X\mbox{ does not pay }Y$$ The contrapositive becomes: $$\forall X,\forall Y\left (X\mbox{ does not pay }Y\Longrightarrow X\mbox{ is not a Lannister or }Y\mbox{ is not a debt of }X \right ) $$ I'm not sure if it's necessary to make the distinction.
Edit: Just to be on the safe side, I would go with the second option. I'm wary of potential wordplay in "A Lannister always pays his debts" and I'm not an English expert.