1.3 Implication

Implication

For two propositions $P$ (the hypothesis) and $Q$ (the conclusion), and some (optional) quantifiers, we can say that $P$ implies $Q$ such that
$(P⟹Q)$

Furthermore, given an implication $P⟹Q$, we can define its

1. ***Converse\neg P) \equiv (\neg P \lor Q)$.
2. Converse: $(Q⟹P)\equiv (\neg Q\vee P)$.

If an implication and its converse are both true, then we say that ”$P$ if and only if $Q$,” such that
$(P⟺Q)\equiv [(P⟹Q)\wedge (Q⟹P)]\equiv [(\neg P\vee Q)\wedge (\neg Q\vee P)]$

Understanding

The implication $P⟹Q$ is the most common form mathematical proofs take. The following are all equivalent ways to say that ”$P$ implies $Q$” in plain English:
$\text{if P, then Q}⟺\text{Q if P}⟺\text{P only if Q}⟺\text{Q is necessary for P.}$
We can also describe the relationship between an implication and its derivatives in a truth table:

\begin{document}
\begin{tikzpicture}
 
% Draw the table borders
\draw (0,0) -- (12,0);
\draw (0,-1) -- (12,-1);
\draw (0,-2) -- (12,-2);
\draw (0,-3) -- (12,-3);
\draw (0,-4) -- (12,-4);
\draw (0,-5) -- (12,-5);
 
\draw (0,0) -- (0,-5);
\draw (2,0) -- (2,-5);
\draw (3.95,0) -- (3.95,-5);
\draw (4.05,0) -- (4.05,-5);
\draw (6,0) -- (6,-5);
\draw (8,0) -- (8,-5);
\draw (10,0) -- (10,-5);
\draw (12,0) -- (12,-5);
 
% Add the table content
\node at (1, -0.5) {$\textbf{P}$};
\node at (3, -0.5) {$\textbf{Q}$};
\node at (5, -0.5) {$P \Longrightarrow Q$};
\node at (7, -0.5) {$Q \Longrightarrow P$};
\node at (9, -0.5) {$Q \Longleftrightarrow P$};
\node at (11, -0.5) {$\neg Q \Longrightarrow \neg P$};
 
\node at (1, -1.5) {T};
\node at (3, -1.5) {T};
\node at (5, -1.5) {T};
\node at (7, -1.5) {T};
\node at (9, -1.5) {T};
\node at (11, -1.5) {T};
 
\node at (1, -2.5) {T};
\node at (3, -2.5) {F};
\node at (5, -2.5) {F};
\node at (7, -2.5) {T};
\node at (9, -2.5) {F};
\node at (11, -2.5) {F};
 
\node at (1, -3.5) {F};
\node at (3, -3.5) {T};
\node at (5, -3.5) {T};
\node at (7, -3.5) {F};
\node at (9, -3.5) {F};
\node at (11, -3.5) {T};
 
\node at (1, -4.5) {F};
\node at (3, -4.5) {F};
\node at (5, -4.5) {T};
\node at (7, -4.5) {T};
\node at (9, -4.5) {T};
\node at (11, -4.5) {T};
 
\end{tikzpicture}
\end{document}

Note that $P⟹Q$ and its contrapositive have the same truth values.

Examples

Implications

Which of the following implications are always true, regardless of $P$? Give a counterexample for each false assertion.

$\forall x\forall yP(x,y)⟹\forall y\forall xP(x,y)$