Relations between two functions

Question

Consider the statements (1) "If $f(i) \geq f(j)$ then $q(i) \geq q(j)$", and (2) "If $q(i) < q(j)$ then $f(i) \leq f(j)$". How can we relate these statements? I mean are these related?

Answer 1

We have : $$(**) \quad(1) \Rightarrow (2)$$

but we have not : $(***) \quad (2) \Rightarrow (1)$

In logic we have :

$$P \Rightarrow Q \Leftrightarrow (\neg Q \Rightarrow \neg P)$$

But her :

$$\neg(f(i \geq f(j)) \Leftrightarrow f(i) < f(j)$$

Since $f(i) < f(j) \Rightarrow f(i) \leq f(j)$ we have only $(**)$ not $(***)$

If $(2)'$ is $q(i) <q(j) \Rightarrow f(i) < f(j)$ then we have $(1) \Leftrightarrow (2)'$

Answer 2

The given statements are actually equivalent with a small modification of the second statement (for a possible typo) viz replacing "$\le$" by "$\lt$" in the second statement.

Let $P$ denote the statement "$f(i)≥f(j)$" and $Q$ denote the statement "$q(i)≥q(j)$"; then $¬P$ would mean "$f(i) < f(j)$" and $\neg Q$ would mean "$q(i)<q(j)$".

We are given (i) $P \Rightarrow Q$ (ii) $\neg Q \Rightarrow \neg P$. The second statement is the contrapositive of the first statement, and hence they are equivalent.