Step 1

Let P be the statement where

\(\displaystyle{P}:{S}{t}{u}{\left.{d}{y}\right.}in{g}\ {h}{a}{r}{d}\ {f}{{or}}\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l}\)

Let Q be the statement, where

\(\displaystyle{Q}:{G}{e}t{in}{g}\ {A}\ in\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l}\) Step 2

Definition: "If A is true then B is true." The logical form of this statement is:

\(\displaystyle{A}\rightarrow{B}\)

The given statement is:

\(\displaystyle\text{If you study hard for your discrete math final you will get A}\)

Note that P: Studying hard for discrete math final and Q: Getting A in discrete math final.

The logical form of the given statement is:

\(\displaystyle{P}\rightarrow{Q}\) Step 3

The given statements is:

\(\displaystyle\text{}{A}ne\ {g}{o}{t}\ {a}{n}\ {A}\ {o}{n}\ {h}{e}{r}\ {d}{i}{s}{c}{r}{e}{t}{e}\ {m}{a}{t}{h}\ {f}in{a}{l};\therefore{J}{a}ne\ mu{s}{t}\ {h}{a}{v}{e}\ {s}{t}{u}{d}{i}{e}{d}\ {h}{a}{r}d\) "Jane got an A on her discrete math final." Hence the statement Q is true for Jane.

This statement implies that "Jane must have studied hard." Hence P is true.

\(\displaystyle{I}{f}\ {Q}{i}{s}\ {t}{r}{u}{e}\ {t}{h}{e}{n}\ {P}{i}{s}\ {t}{r}{u}{e}.\)

The logical form of the statement is:

\(\displaystyle{Q}\rightarrow{P}\) Step 4

From the first statement,

\(\displaystyle{P}\rightarrow{Q}\)

From the second and third statements,

\(\displaystyle{Q}\rightarrow{P}\)

which is not always true.

For example, If the fruit is banana then it is yellow in color; if the fruit is yellow in color one can not assure whether the fruit is a banana.

Hence, converse or inverse error has been made.