Discrete Mathematics - Logic

Question

Could anyone explain what is the correct answer and why are the others are incorrect?

Let P(x) be a property about some object x of type X. If we want to disprove the claim that "P(x) is true for some x of type X", then we have to

1. Show that there exists an x which is not of type X, but for which P(x) is still true.
2. Show that there are no objects x of type X.
3. Show that for every x of type X, there is a y not equal to x for which P(y) is true.
4. Show that for every x of type X, P(x) is false.
5. Show that there exists an x of type X for which P(x) is false.
6. Show that P(x) being true does not necessarily imply that x is of type X.
7. Assume that P(x) is true for every x of type X, and derive a contradiction.

Answer 1

In order to disprove something, all you need to do is provide a counter-example.

For example, to disprove the statement: The Product of two irrationals must be irrational.

a counter example is sqrt(2)*sqrt(8) = sqrt(2*8) = sqrt(16) = 4 which is rational.

So you must show there exists an x of type X for which P(x) is FALSE

That is either option 5.

None of the other statements provide such counterexample...

option #1 is incorrect because the element must be of type X.

option #2 makes the erroneous argument that the statement is false just because the set is empty

option #3 actually supports the statement rather than disproves it

option #6 is the converse of the statement

option #7 is actually the same as option #5. If you assume P(x) holds for all x of type X and

derive a contradiction, then you have shown there exists an x of type X for which P(x) is false.