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11 Jan 2018

3. Negate the following sentence. You may use words or the logical symbols. "For any positive real number 1, there is a natural number n, for which -

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Casey Durgan
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13 Jan 2018
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4. Write the negation of the following statement (you can use words or the logical symbols): "For any two real numbers a, b, if a.b=0 then either a=0 or b=0."

Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parenthesis.

 

Not all precious stones are beautiful. (All stones)

All precious stones are not beautiful. (All stones)

Some isosceles triangle is a right triangle. (All triangles)

No right triangle is isosceles. (All triangles)

Every triangle that is not isosceles is a right triangle.

All people are honest and some people are not honest.

Some people are honest    

Every nonzero real number is positive or negative. (Real number)

Every integer is greater than or less than . (Real number)

Every integer is greater than some integer. (Integers)

No integer is greater than every other integer. (Integers)

Between any integer and any larger integer, there is a real number. (Real number)

There is the smallest positive integer. (Real number)

No one loves everybody. (All people)

Everybody loves someone. (All people)

For every positive real number , there is a unique real number such that . (Real number)

       

a. Given that , find . 

b. Show that, in general, if is a positive real number, then any equation of the form may be written in the form , for some real number . 

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