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Use De Morgan's Law to negate the following statements. a. Vx(x >5) b. 3.(x2+2x + 1 = 0)

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Answer:

a)
\\eg \forall x :x>5 \equiv \forall x:x\leq 5

b)
\\eg [x^2+2x+1=0]\equiv x^2+2x+1\\eq0 or the set
\{x:x\\eq-1\}

Explanation:

First, notice that in both cases we have to sets:

a) is the set of all real numbers which are higher than 5 and in

b) the set is the solution of the equation
x^2+2x+1=0 which is the set
x=-1

De Morgan's Law for set states:


\overline{\rm{A\cup B}} = \overline{\rm{A}} \cap \overline{\rm{B}}\\, being
\overline{\rm{A}} and
\overline{\rm{B}} are the complements of the sets
A and
B.
\cup is the union operation and
\cap the intersection.

Thus for:

a)
\\eg \forall x :x>5 \equiv \forall x:x\leq 5. Notice that
\forall x:x\leq 5 is the complement of the given set.

b)
\\eg [x^2+2x+1=0]\equiv x^2+2x+1\\eq0 which is the set
B = \{x:x\\eq-1\}

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