Grades 11-12 Video Solutions 2021-video 2021 11-12/12

Video Transcription

Problem number 12 states, if A is the union of the open interval from 0 to 1 and the interval from 2 to 3, and B is the union of the open interval from 1 to 2 and the open interval  from 3 to 4, what is the set of all numbers of the form A plus B with A in set A and B in set B? So here we have our two sets and it will be helpful if we see them on the number  line. So A will be the open interval from 0 to 1 and the open interval from 2 to 3. An open interval means that all the numbers are included in the set but not the end points.  So 0.5 is included in the set but not 1. So B will be the open interval from 1 to 2 and the open interval from 3 to 4. And we know that our values lower cases A and B will  fall somewhere on these two intervals. For now let's assume that they will fall on the lower parts of each interval and see what the possible sum could be. To do this we will  add the minimums and maximums together to get a new interval. So in this case we get the open interval from 1 to 3. Now let's move where A is located to the second segment of  A. For the sum we get an open interval from 3 to 5. Now let's move B to the open interval  from 3 to 4 and add them up. We get the open interval from 5 to 7. Let's move back one  for the last combination and we get the open interval from 3 to 5 again. So these are the  intervals where our sum can lie. On the number line this is how they would look. So is this  the open interval from 1 to 7? The answer is no because you can notice that it is impossible  to get the numbers 3 and 5 from the sum. However these numbers would be included in the open interval from 1 to 7. So our final answer is the union of these three intervals.  So the question asked us, what is the set of all numbers of the form A plus B with A in set A and B in set B? The answer is this one, letter D.

Video Summary

The problem involves two sets A and B with defined open intervals. Set A includes (0,1) and (2,3), while set B includes (1,2) and (3,4). The task is to determine possible sums of numbers chosen from each set. By exploring combinations, we find the sums form intervals (1,3), (3,5), and (5,7), excluding numbers 3 and 5. Thus, the numbers formed from A + B are the union of these intervals. The solution clarifies that although the union seems like (1,7), it cannot include numbers 3 and 5, hence the union of these intervals provides the correct answer.

Keywords

set theory

intervals

open intervals

sum of sets

union of intervals