Question number seven. What is the negation of the following statement? Everybody solved more than 20 problems. This statement contains some quantifiers on the participants and then on the number of problems, so let's be absolutely precise in exactly what we mean over here. When we say everybody, everybody means all participants. And then more than 20 means at least 21. We don't know if it's 21 exactly or 25 or all 30 problems, but for sure more than 20, and so we can with certainty say that it must be 21 or more. And now to negate each statement, we would have to say not all participants, which is the same as saying someone. It is not the same as saying none, because we would have to negate each result for each participant, and we are not doing that. So right away we see that choices A and C are not possible. And then to negate that at least 21 problems were solved, we would simply say that less than 21 problems were solved, which is exactly the same as saying less than or equal to 20 problems. So we have three possibilities remaining, and we need a less than or equal to sign with the number 20. So in question D, we have exactly 20 problems, so that's not the right negation. And in E, we have more than 20, which is clearly wrong. So the only remaining choice is less than 21, which we did agree is another possibility. Less than 21 means less than or equal to 20. So we have our answer here, and that is choice B. Somebody solved less than 21 problems.

negation

logical statements

problem-solving

participants

quantifiers