Answer:
Since A and B are not overlapping, their union will simply be the set of all their elements combined. Therefore, a + b will be the sum of the lengths of the two intervals that make up A and B respectively, which is:
a + b = (1-0) + (3-2) + (2-1) + (4-3) = 4
So, a+b=4.
Explanation:
The notation A = ]0,1[ ∪ ]2,3[ means that A is the set of all real numbers that are strictly greater than 0 and strictly less than 1, or that are strictly greater than 2 and strictly less than 3. In other words, A consists of two separate intervals, ]0,1[ and ]2,3[, that do not overlap.
Similarly, the notation B = ]1,2[ ∪ ]3,4] means that B is the set of all real numbers that are strictly greater than 1 and strictly less than 2, or that are greater than 3 and less than or equal to 4. Like A, B consists of two separate intervals that do not overlap.
The union of two sets is the set of all elements that belong to either set. In this case, since A and B do not overlap, their union is simply the set of all real numbers that belong to either A or B. This set can be expressed as the union of the two intervals that make up A and B, which is:
A ∪ B = ]0,1[ ∪ ]2,3[ ∪ ]1,2[ ∪ ]3,4]
To find a + b, we need to add up the lengths of the two intervals that make up A and B respectively. The length of an interval is simply the difference between its upper and lower bounds. So, the length of the interval ]0,1[ is 1 - 0 = 1, the length of the interval ]2,3[ is 3 - 2 = 1, the length of the interval ]1,2[ is 2 - 1 = 1, and the length of the interval ]3,4] is 4 - 3 = 1. Therefore, the sum of the lengths of the two intervals that make up A and B respectively is:
a + b = (1-0) + (3-2) + (2-1) + (4-3) = 4
Hence, a + b = 4.