2

My question is related to the truth table of the conditional operator "implies". i.e. "$r: p \to q$".

\begin{array}{c c | c} p & q & p \to q\\ \hline T & T & T\\ T & F & F\\ F & T & T\\ F & F & T \end{array}

Consider this question in an example:

Let $Q(x,y)$ be the predicate "If $x<y$, then $x^2<y^2$" with both $x$ and $y$ real numbers. Find the truth set of $Q(x,y)$.

Should I consider the cases where "$x>y$" and "$x=y$"? Should these pairs be included in the truth set? It confuses me because according to the truth table or definition of conditional operator, the statement "If p, then q" is true when the proposition p is false. However to my mathematical intuition whenever we provide a condition, it means we are talking about q assuming p is correct.

N. F. Taussig
  • 79,074
CathIAS
  • 125

1 Answers1

2

EDITED

The truth set is

$$\{(x,y):x<y,|x|\ge|y|\}^c=\left[\{(x,y):x<y\}\cap \{(x,y):|x|\ge|y|\} \right]^c=$$ $$= \{(x,y):x\ge y\}\cup \{(x,y):|x|<|y|\}.$$

That is, the truth set is the complement of the set where the implication is false according to the truth table. ($p(x,y)$ is true and $q(x,y)$ is false; $x<y$ and $x^2\ge y^2$.)

It may be surprising that the truth set is simply the union of the sets where $p(x,y)$ is false and $q(x,y)$ is true.

It is easy to check all the possible combinations in the figure below. (Color red indicates that the premise is false, that is, there is no need to further investigation.)

enter image description here

If this example goes against your intuition then think of the following implication: Let $R=$ "it is raining." $W=$ "the grass is wet".

$$R \rightarrow W.$$

Is this false just because it is not raining right now?

It is worth to take a look at this question. (You will find there further explanation regarding the naive set theoretic version of implication of $A \rightarrow B=A^c\cup B$)

Community
  • 1
zoli
  • 20,817
  • How about the case where x>y>0? In this case both antecedent and consequence are false, so the whole statement is true? – CathIAS Sep 11 '15 at 09:04
  • @CathIAS: Formally (see the truth table): certainly. Regarding my example: if it is not raining and the garden is wet.... (My sprinkler is working.) Does that mean that the original implication is false? – zoli Sep 11 '15 at 09:12
  • then why the truth set doesn't include that part? – CathIAS Sep 11 '15 at 09:17
  • @CathIAS: YOU WIN: I edit. – zoli Sep 11 '15 at 09:21
  • @MauroALLEGRANZA: For $(x,y)=(2,4)$ both $2<4$ and $|2|<|4|$ hold. Also, $2^2<4^2$ holds. So the implication is true. SO I don't completely understand the question. – zoli Sep 11 '15 at 10:25
  • @MauroALLEGRANZA: I've edited my answer bacause I did not like it as it was. (Acceptance does not mean that I fall in love with my answer.) – zoli Sep 11 '15 at 11:57