8. Use De Morgan’s laws to find the negation of each of the following statements.

De Morgan’s Laws:

¬(p∨q)≡¬p∧¬q

¬(p∧q)≡¬p∨¬q

As in a previous exercise, if we use propositional variables, we will not make mistakes while solving this type of exercise.

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a) Kwame will take a job in industry or go to graduate school

Let p=”Kwame will take a job in industry” and q=”Kwame will go to graduate school”.

¬(p∨q)≡¬p∧¬q, by the De Morgan Law.

Therefore, the negation of this statement is Kwame won’t take a job in industry, and Kwame won’t go to graduate school.

b) Yoshiko knows Java and calculus

Let p=”Yoshiko knows Java” and q=”Yoshiko knows calculus”.

¬(p∧q)≡¬p∨¬q, by the De Morgan Law.

Therefore, the negation of this statement is Yoshiko doesn’t know Java, or Yoshiko doesn’t know calculus.

c) James is young and strong

Let p=”James is young” and q=”James is strong”.

¬(p∧q)≡¬p∨¬q, by the De Morgan Law.

Therefore, the negation of this statement is James is not young, or James is not strong.

d) Rita will move to Oregon or Washington

Let p=”Rita will move to Oregon” and q=”Rita will move to Washington”.

¬(p∨q)≡¬p∧¬q, by the De Morgan Law.

Therefore, the negation of this statement is Rita won’t move to Oregon, and Rita won’t move to Washington.

Note: When you are negating a compound proposition, always substitute the statements with propositional variables to avoid mistakes.

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