Question

s(3+3 marks) Write down the negations of the following expressions, so that the symbol does not arise to the left of any quantifier. Indicate whether the negated statement is true. 2x

Answer 1

5.The rule for the negation is belongs to becomes not belongs to and not belongs to becomes belongs to . Also or becomes and , and becomes or . Also for all becomes there exist

a ) The negation of the given statement is ,

The rule for the negation is belongs to becomes not belongs to and not belongs to becomes belongs to. Also or becomes and, and becomes or . Also for all becomes there exist The negation of the given statement is, for all x euro N, x^2 NotElement N or 1/2x euro N. This negated statement is not a true statement, for counter example 1 euro N and 1^2 = 1 euro N so x^2 NotElement N is not true . The negation of the given statement is, For all x euro R, for all n euro Z, x^n lessthanorequalto 0. This negated statement is also not true. For counter example take x = 2 and n = 3 then x euro R and n euro Z but x^n = 2^3 = 8 > 0 u.e., x^n lessthanorequalto 0 is not true so the statement is not true.
for all x € N , x²∉ N or $\frac{1}{2x}$ € N .

This negated statement is not a true statement , for counter example 1 € N and 1²=1€ N so x² ∉ N is not true .

b) The negation of the given statement is ,

For all x € R , for all n€Z , xⁿ$\leq$ 0.

This negated statement is also not true . For counter example take x= 2 and n=3 then x€ R and n€Z but xⁿ= 2³=8 >0 i.e., xⁿ$\leq$ 0 is not true so the statement is not true.

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If you have any doubt or need more clarification at any step please comment.