EXAMPLE 2.2.8

1. Select the statement that is the negation of "If you know the password, then you can get in."

A. If you don't know the password, then you can get in.

B. You don't know the password or you can get in.

C. You don't know the password and you can't get in.

D. You know the password and you can't get in.

SOLUTION

Let p be "You know the password." Let q be "You can get in." The statement "If you know the password, then you can get in" has the form "If p, then q." We are asked to select the negation, which has the form "p and not q."

The correct choice is "D. You know the password and you can't get in."

2. Select the statement that is logically equivalent to "If you pass MGF1106, then a liberal studies math requirement is fulfilled."

A. If a liberal studies math requirement is fulfilled, then you passed MGF1106.

B. You pass MGF1106 and a liberal studies math requirement is fulfilled.

C. You don't pass MGF1106, or a liberal studies math requirement is fulfilled.

D. You pass MGF1106, or a liberal studies math requirement is not fulfilled.

SOLUTION

Let p be the statement "You pass MGF1106." Let q be the statement "A liberal studies requirement is fulfilled." The statement "If you pass MGF1106, then a liberal studies math requirement is fulfilled" has the form "If p, then q." We are asked to select an equivalent statement. We know that an equivalent form to "If p, then q" is "Not p, or q." Applying this rule we see that the correct answer is "C. You don't pass MGF1106, or a liberal studies math requirement is fulfilled."

3. Select the statement that is the negation of "If you have income from royalties, then you must complete Schedule E."

A. You have income from royalties and you must complete Schedule E.

B. You have income from royalties and you don't have to complete Schedule E.

C. You have income from royalties or you must complete Schedule E.

D. You have income from royalties or you don't have to complete Schedule E.

SOLUTION

Let p be "You have income from royalties." Let q be "You must complete Schedule E." The statement "If you have income from royalties, then you must complete Schedule E" has the form "If p, then q." We are asked to select the negation, which has the form "p and not q." Applying this rule we see that the correct choice is "B. You have income from royalties and you don't have to complete Schedule E."