Question 1109244

demorgan's law states:


not (A OR B) = not (A) AND not (B).


not(A AND B) = not (A) OR not (B).


your original statement is:


(x <= -1 OR x > 1)


the demorgan law that applies is:


not (A OR B) = not (A) AND not (B).


if you let A = (x <= -1) and B = (x > 1), then you get:


not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1).


not(x <= -1) becomes (x > -1)


not(x > 1) becomes (x <= 1)


your equation of:


not((x <= -1) OR (x > 1)) = not (x <= -1) AND not (x > 1) becomes:


not((x <= -1) OR (x > 1)) = (x > -1) AND (x <= 1).


x > -1 is the same as -1 < x.


therefore (x > -1) AND (x <= 1) becomes:


(-1 < x) AND (x <= 1).


this can be written as -1 < x <= 1.


that would be the same as the second selection.


your questions are:


question 1:


Why do the 2nd and 3rd answers not include the word AND? Is there a significance? 


the 2d answer is -1 < x <= 1


that is the same as saying that -1 < x AND x <= 1.


the 3d answer is -1 <= x < 1.


that is the same as saying that -1 <= x AND x < 1.


the AND, in either one of these cases, is unnecessary.


question 2:


I thought that to negate an equality sign, you merely flipped it. > becomes <, and <= becomes >=. 


you are probably thinking of what happens when you multiply both sides of an inequality by a negative number.


if your inequality is x < y and you multiply both sides by -1, then you get -x > -y.


if your inequality is x <= y and you multiply both sides by -1, then you get -x >= -y


in both cases, you are reversing the inequality.


that is not the same as negating the inequality itself.


a negation of x < y means x >= y.


a negation of x <= y means x > y.


a negation of x > y means x <= y.


a negation of x >= y means x < y.


if x is smaller than y, then x cannot be equal to y nor can it be greater than y.


therefore, the negation of x smaller than y means that x can either be greater than y or equal to y.


if x is greater than y or equal to y, x cannot be smaller than y.