CISE Necd exercises 1.8.s and 1.8.1 Negate the following statements. Write the n
ID: 3169321 • Letter: C
Question
CISE Necd exercises 1.8.s and 1.8.1 Negate the following statements. Write the negation as a positive statement,to whatever extent is possible. 1. There exists a line in the plane passing through the points (-1. 1D), (2, -1), and 2. There exists an odd prime number. 3. For all real numbers x, x3- 4. Every positive integer is the sum of distinct powers of three. 5. For all positive real numbers x there exists a real number y such that y2-x. 6·There exists a positive real number y such that for all real numbers x, y2-x. 1.8.6 EXAMPLE Sometimes it helps to deal with very complex statements more carefully, one step at a time. Consider the statement "All Martians are short and bald, or my name isn't Darth Vader." We now consider various substatements: *A := All Martians are short and bald. ·B := My name isn't Darth Vader. C: All Martians are short. e D : All Martians are bald. Clearly, A is equivalent to C A D. Our original statement is A v B. So the negation of our original statement is We can now clearly see that the negation of "All Martians are short and bald or my name isn't Darth Vader" is "Either some Martian is tall or some Martian has hair, and my name is Darth Vader." 1.8.7 PROBLEM If you want a challenge, try using this process to negate You can fool some of the people all of the time, and some of the people none o the time, but you cannot fool all of the people all of the time.Explanation / Answer
1.8.5
1.) For every line L in the plane, L does not pass through the points (-1,1) (2,-1) and (3,0).
2.) For all odd numbers x, x is not prime.
3.) There exists a real number x such that x3 x.
4.) There exists a positive integer such that it is not the sum of distinct power of three.
5.) There exists a positive real number x for all real numbers y such that y2 x.
6.) For all positive real number y, there exists a real number x such that y2 x.