Math 2.0

definition

Subset

Definition

Let AA and BB be sets. The set AA is a subset of BB, written A⊆BA \subseteq B, when each element of AA is an element of BB: ∀x (x∈A→x∈B).\forall x \, (x \in A \to x \in B). Then BB is a superset of AA. The set AA is a proper subset of BB, written A⊊BA \subsetneq B, when A⊆BA \subseteq B and A≠BA \neq B.

Remarks

Some authors write A⊂BA \subset B for A⊆BA \subseteq B, and others write it for a proper subset. Two sets are equal if and only if each is a subset of the other. This follows from the axiom of extensionality. The empty set is a subset of every set, and every set is a subset of itself.

Explanations

L1 human

A⊆BA \subseteq B means that AA fits inside BB. The set of even numbers is a subset of the set of integers. To prove A⊆BA \subseteq B, take an arbitrary x∈Ax \in A and show x∈Bx \in B.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Subset, lead section

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