Question

1. Convert the following statement to a quantified expression, negate it, and convert the answer back to English. You will end up with the exact opposite concept.

All cats are asleep.

2. Convert the following statement into a quantified expression:

Everyone, who has seen Deadpool, likes chimichangas .

3. Simplify the following Quantified Statement. The result should have no negation symbols. ¬ ?x ?x (¬G(x) ? H(x) )

4. Prove the following using induction (show your work- both steps):

If x?2then2+4+6+...+2n = n(n+1)

Answer 1

SOLUTION TO QUESTION 1:

Let C(x) denote that x is a cat and S(x) denote that x is asleep.

Hence, ALL CATS ARE ASLEEP can be written as:

$\forall x [C(x) \rightarrow S(x)]$

The negation of this statement is given as

$\neg \forall x [C(x) \rightarrow S(x)]$   

which is equivalemnt to

$\exists x \neg [C(x) \rightarrow S(x)]$   

which can be interpreted as -

There exists an element x for which x is asleep does not hold. In other words, there is at least a cat which is not asleep.

SOLUTION TO PROBLEM 2:

Everyone, who has seen Deadpool, likes chimichangas.

Assume that D(x) means that x has seen Deadpool and C(x) implies that x like chimichangas. Thus, the above statement can be re-written in quantifier form as:

$\forall x [D(x) \rightarrow C(x)]$

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The problem 3 and 4 are beyond understanding. I guess the question replaces some important quantifier/implication condition with ? sign which cannot be interpreted here. I will advise you to re-post these questions with correct conditions.