How to figure out the contrapositive and negation of statements.

How do I figure out what the contrapositive and negation of the two statements below are? Could you also give a way of finding the negation and contrapositive for any statement in simple terms.

(Pâˆ¨Q)â‡′آ¬R(P âˆ¨ Q) â‡’ آ¬R

Pâ‡′(Qâ‡′R)P â‡’ (Q â‡’ R)

=================

Hint: Find the ‘outermost’ implication, then reverse it and negate the statements on both sides of that implication. By outermost, I mean the one that would be applied last according to the order of operations.
– Justin Benfield
2 days ago

=================

2

=================

Following my hint:

(P∨Q)⇒¬R(P\lor Q)\Rightarrow\neg R

Contrapositive is:

¬(P∨Q)⇐¬(¬(R))\neg (P\lor Q)\Leftarrow\neg(\neg(R))

Which simplifies to:

((¬P)∧(¬Q))⇐R((\neg P)\land(\neg Q))\Leftarrow R or in the usual order: R⇒((¬P)∧(¬Q))R\Rightarrow((\neg P)\land(\neg Q)).

For P⇒(Q⇒R)P\Rightarrow(Q\Rightarrow R), the leftmost ⇒\Rightarrow is the outermost, so:

¬P⇐¬(Q⇒R)\neg P\Leftarrow\neg(Q\Rightarrow R)

Which simplifies to

¬P⇐(Q∧¬R)\neg P\Leftarrow(Q\land\neg R) or in the usual order: (Q∧¬R)⇒¬P(Q\land\neg R)\Rightarrow\neg P.

So is the contrapositive like taking the not of both sides
– Jaimy Lunn
2 days ago

And reversing the sign
– Jaimy Lunn
2 days ago

More or less, specifically, given an implication, P⇒QP\Rightarrow Q, the contrapositive statement is (¬Q)⇒(¬P)(\neg Q)\Rightarrow(\neg P). It does not matter what PP and QQ are, they can be entire logical expressions or merely ‘primitives’ (which have only a truth value). The point is: If the implication holds, so to does the contrapositive implication.
– Justin Benfield
2 days ago

Ok thanks but how would I figure out the negation of each statement? Sorry 🙁
– Jaimy Lunn
2 days ago

That’s an exercise is basic formal logic, are you familiar with the truth tables for the logical operations of ¬\neg, ∧\land, ∨\lor, ⇒\Rightarrow, ⇔\Leftrightarrow?
– Justin Benfield
2 days ago

The contrapositive of p⇒qp\Rightarrow q is, as has already been pointed out, ¬q⇒¬p\lnot q \Rightarrow \lnot p. (This actually means the same thing as p⇒qp\Rightarrow q, as can be verified using truth tables or logic algebra).

For the negations of the propositions you presented, you might find the basic logic equivalences useful (see a list at the Wikipedia article). Of particular interest in negating implications are the equivalences (both of which can be proved exhaustively using truth tables):

p⇒q≡¬p∨qp\Rightarrow q \equiv \lnot p \lor q

¬(p∨q)≡¬p∧¬q\lnot (p\lor q) \equiv \lnot p \land \lnot q

It also depends on what we would like to express the negation in terms of. For example, do we want to express the negation only in terms of negations, disjunctions and conjunctions? Do we want the negation expressed as an implication?

An example of an expression of the negation P\lor Q \Rightarrow \lnot RP\lor Q \Rightarrow \lnot R:

\begin{align}
\lnot\left(P\lor Q \Rightarrow \lnot R\right) &\equiv \lnot\left(\lnot(P\lor Q) \lor \lnot R\right)\\
&\equiv (P\lor Q)\land R
\end{align}\begin{align}
\lnot\left(P\lor Q \Rightarrow \lnot R\right) &\equiv \lnot\left(\lnot(P\lor Q) \lor \lnot R\right)\\
&\equiv (P\lor Q)\land R
\end{align}