Bruno,
Can you provide definitions of "belongs-to" and "included-in" that
distinguish them from "union" and "intersection"?

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Here we met a set of sets.
The set of subsets of a set, can only be, of course, a set of sets. The set
{2, 21, 14} is a set of numbers. The set { { }, {4, 78, 56} } is a set of sets.
It has two elements: the empty set {}, and the set of numbers {4, 78, 56}. Do
not confuse a number, like 24, and a set, like {24}, which is a set having a
number has elements. In particular it is the case that {4, 78, 56} belongs to
{ { }, {4, 78, 56} }. Take it easy, and meditate on the following exercise:
Which of the following are true
{3, 5} included-in {3, 5} True
{3, 5} belongs-to {3, 5}
{3, 5} included-in { {3, 5} }
{3, 5} belongs-to { {3, 5} }
Take your time,
Bruno
http://iridia.ulb.ac.be/~marchal/
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