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Complements in set theory

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Let's look at complements in set theory. The "relative complement" is two sets is the difference between the two. As in ordinary arithmetic, we represent the difference between two sets with the minus sign. A - B is the difference between sets A and B. In set theory, we would write A - B = {x ∈ A: x ∈' B}. We can also represent the relative complement of any two sets by writing A \ B. We can thus represent the complement with the apostrophe. We would describe the symbols in the set builder notation as stating that "x is an element of A is such that x is not a member of B." It's in this sense that the relative complement of one set is precisely what is not in the other set.

De Morgan's laws apply to the relation of complements to both unions and intersections.

Thus:

(A ∪ B)' = A' ∩ B' is true as well as (A ∩ B)' = A' ∪ B'

Halmos also speaks of the "symmetric difference," also known as the "Boolean sum" of two sets. The symmetric difference or Boolean sum of two sets is the union of the difference and the inverse of that difference of two sets. Thus, we can represent it as follows:

A + B = (A - B) ∪ (B - A)

In addition to writing A + B, we can represent the symmetric difference or Boolean sum of any two sets by writing A ∆ B

Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.

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