Bifalgorithm post · Brisk · Chapter 1

How can a set be an element of another set without being its subset?

How can a set be an element of another set without being its subset? (Brisk · Chapter 1) — "Let’s give some examples. Suppose that Q is the set {4, {9, 4}, 2}.…

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Exemples de relations entre ensembles, appartenance et inclusion
Définition des ensembles utilisés
Q = {4, {9, 4}, 2} possède trois éléments, dont un ensemble.
P = {4, 9}.
R = {2, 4}.
Appartenance de P à Q
P ∈ Q est vrai.
P est le même ensemble que {9, 4}, car l’ordre des éléments ne compte pas.
Inclusion de P dans Q
P ⊆ Q est faux.
Pour avoir P ⊆ Q, chaque élément de P devrait appartenir à Q.
Cette condition échoue, car 9 n’appartient pas à Q.
R ⊆ Q est vrai, car 2 et 4 appartiennent tous deux à Q.

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How can a set be an element of another set without being its subset?

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Brisk

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"Let’s give some examples. Suppose that Q is the set {4, {9, 4}, 2}. Q has three elements here, one of which is itself a set. Now suppose that we let P be the set {4, 9}. Question: is P ∈ Q? The answer is yes: the set {4, 9} (which is the same as the set {9, 4}, just written a different way) is in fact an element of the set Q. Next question: is P ⊆ Q? The answer is no, P is not a subset of Q. If P were a subset of Q, that would imply that every member of P (there are two of them: 9 and 4) is also an element of Q, whereas in fact, only 4 is a member of Q, not 9. Last question: if R is defined to be {2, 4}, is R ⊆ Q? The answer is yes, since both 2 and 4 are also members of Q."

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