Answer to Question #308960 in Discrete Mathematics for Zain

Question

Determine if the following argument is valid or if it exhibits the converse or the inverse error. Use symbols to write the logical form of argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made.

If at least one of these two numbers is divisible by 6,

then the product of these two numbers is divisible by 6.

Neither of these two numbers is divisible by 6.

∴ The product of these two numbers is not divisible by 6.

Answer 1

Let the statements be:

p = "at least one of these two numbers is divisible by 6"

q = "the product of these two numbers is divisible by 6"

Then the statement "Neither of these two numbers is divisible by 6" is -p and the statement "The product of these two numbers is not divisible by 6" is -q. Thus, the argument has the following form:

$$p\Rightarrow q\\ -p\\ \therefore -q$$

This is inverse error.