In details

The fifth truth


How can we make "intuitions"of union, intersection, difference, complement, and symmetric difference, mathematical concepts? Let us begin with the intersection. We define the intersection set of A and B as the set of sets belonging to A and B simultaneously. By which theory authorizes us to give this definition is that the given sets A and B form a nonempty set D which we will write as follows: D = {A, B}. By Axiom 2, the axiom of subsets, we can say that there is the set of x such that it "belongs to A and B simultaneously" because that property refers to the sets of the nonempty set D. Thus, we are authorized by Axiom 2 to assert the existence of the intersecting set of two sets A and B. Similarly we can argue that given a set of sets D, there is the set of sets x that belong to all sets of D. In short, given two sets A and B, there is the set

A Ç B = {x: x belongs to A and x belongs to B}.

In order to define the assembly of sets A and B we cannot proceed in the same way. That is, we cannot demonstrate that there is the assemblage set of A and B from the four axioms we have so far (axioms 0, 1, 2, and 3). We need a new axiom: Axiom 4, called the Reunion Axiom.

Axiom 4

For every set C, there is a set U such that

if x belongs to M, for some M that belongs to C, then x belongs to U.

Put another way, given a set of sets C, there is the set of sets that belong to some set of C. We can still read this axiom in other ways. For example, we can say that there is the set of sets belonging to the sets of C for any given set C. For example again, we can say that given the sets A and B, there is the set of sets belonging to A and B simultaneously. In this case we first form the set C = {A, B} and then we form the set U of the sets that belong to A or B. That is, we write: U = A È B.

We now have five axioms and the most recent of them allows us to form the assembly set. With the meeting axiom we can form the "tender" set, generalizing the concept of the "even" set. Given sets A, B, and C, we define, with the help of the meeting axiom, set {A, B, C} as the set of sets {A}, {B}, and {C}. Notice that the set {A} exists because of the pair axiom that says {A, A} is set. That is, {A, A} = {A} is a new set. Similarly, sets {B} and {C} also exist, and therefore, by the meeting axiom we can form the assembly set {A} È {B} È {C} = {A, B, C}.

It is interesting to note that to get the two or more sets together we need a new axiom, the meeting axiom. We suggest that you give some thought to the need for this new axiom. Try to think about how it would be possible to conceive of the assembly without a new "truth" being "invented" so as not to be questioned.

The complement of B to A is easy to define: A - B = {x: x belongs to A but does not belong to B}. We can also say that A - B is the difference between A and B. Finally, the symmetrical difference between A and B is defined by: A D B = (A - B) È (B - A).

Challenge for you: Be convinced that the complementary, the difference, and the symmetrical difference do not require new axioms.

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