Question

Use symbols to write the logical form of the following arguments. If valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error has been made. If you study hard for your discrete math final you will get an A. Jane got an A on her discrete math final. Therefore, Jane must have studied hard.

Discrete math
ANSWERED
asked 2021-08-03
Use symbols to write the logical form of the following arguments. If valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error has been made.
If you study hard for your discrete math final you will get an A.
Jane got an A on her discrete math final.
Therefore, Jane must have studied hard.

Answers (1)

2021-08-04
Let P be the statement where
P: Studying hard for discrete math final
Let Q be the statement, where
Q: Getting A in discrete math final
Definition: "If A is true then B is true." The logical form of this statement is:
\(\displaystyle{A}\rightarrow{B}\)
The given statement is:
"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:
\(\displaystyle{P}\rightarrow{Q}\)
The given statements is:
"Jane got an A on her discrete math final. therefore Jane must have studied hard."
"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.
If Q is true then P is true.
The logical form of the statement is:
\(\displaystyle{Q}\rightarrow{P}\)
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.
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