Apr 16, 2020 · A contrapositive is a statement that says that if P is true, then Q is true. The opposite must also be true, meaning if Q is false, P has to be false. Therefore, the contrapositive has to be "For all dogs A,B,and C, if B AND C are not shibas, A OR B are not male" Check De Morgan's Law for your reference

A statement like "if all x are P, then some y is Q" would be" if there does not exist a y that is Q, then it is not the case that all x are P$. But that isn't really the case here. $\endgroup$ – amWhy

A contrapositive has truth value equivalent to the original statement: $$\text{It is raining}\implies\text{I have an umbrella}$$ has a contrapositive (and is equivalent to) $$\text{I do not have an umbrella}\implies\text{it is not raining}$$ Proving the contrapositive is equivalent to proving the original statement, and can sometimes be cleaner.

Jan 19, 2018 · To see that, consider the original statement itself (if the statement is true, so is the contrapositive). This is one of the cases where the truth of the contrapositive is less obvious than the truth of the original statement.

So basically, in a proof by contrapositive, you assume that ~C is true, and prove that when ~C is true, it leads to ~A and ~B. That is pretty much all you need to do. So in response to your question, in a proof by contrapositive it does not matter whether you assume that A and B …

Jul 30, 2019 · No, that'd be the inverse; the contrapositive here is ~r → ~(p ∧ q). Note that not only did I negate both sides, I switched the order. For some intuition, try considering the contrapositive of the plain English statement "If this is a rose, it is a flower".

$\begingroup$ It also happens, one tries to argue by contradiction or contraposition when the negation is simpler than the assertion, for example, if an $\forall$ gets turned into a $\exists$, we can work with something less restrictive.

Prove the following statement by proving its contrapositive: “If $n^3 + 2n + 1$ is odd then n is even” Therefore: $\lnot q \rightarrow \lnot p =$ "if $n^3 + 2n ...

I converted this example into logical notation with quantifiers, which makes the difference between negation and contrapositive more obvious. Original statement: Any day when Mr. So-and-so is happy is a day when he sings. $$\forall d \in D_{ays}: (H_{appy}(M_r) \rightarrow S_{ings}(M_r))$$ Contrapositive: Any day when Mr. So-and-so does not ...

Dec 28, 2019 · I want to understand contrapositive clearly. I'll start by saying: "If it is sunny, then there is light". The statement is true. But now consider the contrapositive: " If there is no light then it is not sunny". The contrapositive is false because you could create light with a …