New comments in italics.
For example {1,2} INTERSECTION {2, 7} is equal to some set, actually the set
{2}. OK?......No!
Why not the sets {1,2,7} if INTERSECTION means
BOTH?
Ah, but the word "both" alone is ambiguous. You could say that the UNION of
two sets is the merging of BOTH set, and the intersection is the given of the
elements which are in both set. So the union of {1, 2} and {2, 7} is {1, 2, 7},
which indeed merges BOTH sets. But for computing the intersection, you must ask
yourself, does this *element* belongs to BOTH set? So, for the intersection of
{1, 2} and {2, 7}, you have to ask yourself the following question: does 1
belong to both set? well, the answer is NO. the 1 belongs to the first set but
not to the second, and so 1 does not belong to the intersection. Does 2 belongs
to both sets? The answer is yes. 2 belongs to {1, 2} and 2 belongs to {2, 7}.
Does 7 belongs to both sets, the answer is no, 7 belongs to the second set, but
does not belong to the first set, so 7 is not in the intersection.
Tell me if you are OK with this.
Not OK. You previously
defined UNION as one OR the other. Now you seem to be giving me the same
definition for INTERSECTION.
I'm back! I give you two last exercises to ponder about, just in case
of insomnia. Again, take your time. I hope Kim follows, and does not look at
the solution !
1°) In the two relational formula below, one is true, the other is false.
Which one are what?
a) { } INCLUDED-IN { }True
Very good. All elements of { } are among the elements of { }. This is
sometimes said to be "trivially" true, because { } has no elements at all.
This is an example of an important "fine point". Examples:
To verify if the set {1, 2, 3} is included in {34, 56, 7, 2, 100, 1, 45, 3,
4}, you have to check THREE things: does 1 belongs to the second set, does 2
belongs to the second set, does 3 belongs to the second set.
To verify if the set {1, 2} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4},
you have to check TWO things: does 1 belongs to the second set, does 2 belongs
to the second set.
To verify if {1} is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to
check ONE things: does 1 belongs to the second set.
To verify if { } is included in {34, 56, 7, 2, 100, 1, 45, 3, 4}, you have to
verify ZERO thing! So it is automatically true. That is why logicians say it is
trivially true.
From this you should understand that the empty set, { }, is included to any
set.
So { } is included in all the sets: { }, {1}, {1, 2}, ... {0, 1, 2, ...},
....
In particular, as you said correctly, { } is included in { }. Put in another
way, ({ } INCLUDED-IN { }) = true.
b) { } BELONGS-TO { } True
NOT correct. Remember that the empty set is empty, so nothing belongs to it.
All formula like (x belongs to { }) will be false. You can conceive a set as an
empty box { }, in which you can fill elements. So the set {a, b} is the empty
set in which you put the elements a, and then, the element b. The accolades "{"
and "}" represents the box itself, and what is in between the accolades
represents the elements of the set. You could have guessed the solution because
I was helping you when saying that one of the proposition is true and the other
is false, and this means that, like many beginners, you read the enunciation of
the problem too much quickly. That is why I suggest you take your time, and
read often, at different time, the enunciation of the problems, and actually
all explanations as well.
The moral is:
"x belongs to { }" is never true, or is always false, whatever x
represents.
"{ } included in x" is always true, or never false, whatever x represents.
2°) And I give you a slightly longer exercise. Can you give me all the
subsets of the set {1, 2} ?. That is, can you give me all the sets which are
included in the set {1, 2} ? In case of doubt, reread the definitions, reread
the examples, and never panic! I give you a hint: the set {1, 2} has four
subsets. Can you find them?
{1} {2} {1,2} {2,1} why not {3} ?
Not too bad. 3/4 correct:
{1} is included in {1, 2}. Indeed.
{2} is included in {1, 2}. Indeed.
{1, 2} is included in {1, 2}. Indeed.
{2, 1} is included in {1, 2}. Indeed, that is true, but you have to remember
what you have already agree on: the set {1, 2} is equal to the set {2, 1}, so
this is not a new solution. It is the preceding one in disguised!
Why not {3}? {3} is not included in {1, 2} just because 3 does not belong to
{1, 2}. Reread the definition of inclusion. A is included in B if all the
elements of A belongs to B. OK?
So you have found three subsets, among the four. Reading today's explanations
I think you could find the missing subset. I let you search a little bit.
So just one exercise: what is the missing subset?
Is the missing subset { } ?
I apologize if all of this is a bit boring, but soon enough it will be highly
rewarding. You will see.
Bruno
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