For those beginning to study mathematics, a seemingly very difficult question to answer is as follows: How do I know what the next truth I can demonstrate? Over 2500 years ago Greek mathematicians discovered that "human intuition"is one of the main resources in mathematical research, and that it serves as"guide of human reason"Euclid used figures to draw inspiration from his geometric discoveries and to guide his demonstrations.
Let's use our "topological (space) intuition" to inspire our investigation of ensembles. Let's start with a picture.
How many regions can we identify in the figure above? We immediately noticed fifteen interesting regions: (1) The white plus yellow, blue and green regions together form the "universe". (2) The green region. (3) The white, yellow and green regions. (4) The white, blue and green regions. (5) The yellow, blue and green regions. (6) The white and yellow regions. (7) The white and blue regions (8) The white and green regions (9) The yellow and blue regions (10) The yellow and green regions (11) The blue and green regions (12) A white region. (13) The yellow region. (14) The blue region. (15) The green region.
How can we make these "intuitions"Mathematical concepts? How can we describe these regions as sets? We can start with a set that contains all the others, that is, we define the universe set U. To define the other sets we will be guided by"intuition"of the figures. Our first task is to define the"union of two sets", then we'll define the"intersection of two sets"then the"complementary to a set"then the"two sets difference"then the"symmetrical difference of two sets".
The yellow region plus the blue region plus the green region form the whole of A and B: we'll write AÈ B. The blue region is the part that is in both set A and set B: we write AÇ B. The yellow region is the part from set A that is not in set B, we write AB, and the green region is the part of set B that is not in set A, and we write BA. Each of these two sets we call "difference"between two sets. We can see A - B as the"complementary set of B in A", that is, the set of sets that belong to A but do not belong to B. We will then say that U - A is the"complementary set of A"that is, it is the set of sets of the universe that do not belong to A. If we unite the differences between A and B and between B and A, we have the"symmetrical difference of A and B", which will be written by the symbol A D B.
Our big problem right now is to show that all these definitions are allowed by the axioms. In other words, we need to show that our "intuition"can be perfectly represented by"theory"that we have so far.
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