Use the facts that the negation of a statement is a statement and that the negat

Question

Use the facts that the negation of a statement is a statement and that the negation of an if-then statement is an and statement to rewrite each of the statements without using the word necessary or sufficient. Show work and steps require to get the answer.

a) Being divisible by 8 is not a necessary condition for being divisible by 4.
b) Having a large income is not a necessary condition for a person to be happy.
c) Having a large income is not a sufficient condition for a person to be happy.
d) Being a polynomial is not a sufficient condition for a func- tion to have a real root.

Explanation / Answer

a) Being divisible by 8 is not a necessary condition for being divisible by 4”.

If we let P(x) to be the predicate “x is divisible by 4” and Q(x) to be the predicate “x is divisible by 8”, then the given statement can be written as:

(P(x)Q(x))x(P(x)Q(x))

Some integers are divisible by 4 and are not divisible by 8

b)    Denote by P(x) the predicate "Having a large income"

and Q(x) the predicate "person to be happy"

      P= a large income
o   Q=a person to be happy
o   Quantifier= a person, There exists a person, x
o   x , P /> Q = x, ~(P --> Q )
·         Now, simplify and translate into English
o   x, P~Q
o   Answer: There exists a person who has a large income and is not happy

c) Identify quantifiers, P and Q.
o   Quantifier= Being, which is another way to interpret x.
o   P= a large income
o   Q= a person to be happy
o   x, P /> Q
·         Rewrite statement,
o   x, ~(P àQ)= x, P~Q

d) Being a polynomial is not a sufficient condition for a function to have a real root.
·         Identify quantifiers, P and Q.
o   Quantifier= Being, which is another way to interpret x.
o   P= a function that is a polynomial
o   Q= to have a real root
o   x, P /> Q
·         Rewrite statement,
o   x, ~(P àQ)= x, P~Q
o   So in an English sentence this can be understood as: There exists a function that is a polynomial that does not have a real root.

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