📂Set Theory

Set Inclusion

Definition ¹

$$ A \subset B \iff \forall x (x\in A \implies x \in B) $$ For any set $A$, $B$, if all elements of $A$ are also elements of $B$, then $A$ is a Subset of $B$, and $B$ is a Superset of $A$, denoted as $A \subset B$.

Explanation

If it’s $A \subset B$ and $B \not\subset A$, then $A$ is called a Proper Subset of $B$, and denoted as $A \subsetneq B$.

As a minor note, $A \subset B$ means $A$ is included in $B$, and $a \in A$ means $a$ belongs to $A$. Although it might seem the same to many, there are often confusions in actual language habits, and most people won’t fuss over it as long as they understand each other. However, inclusion is a relation defined between sets, and $a \in A$ is not referred to as ’the belonging relation of a set and an element’, which is a difference worth knowing.

Theorem: Transitivity of Inclusion

For any set $A$, $B$, $C$, $$A \subset B \land B \subset C \implies A \subset C$$

Proof

According to the assumption, $$ A \subset B \iff \forall x (x\in A \implies x \in B) \\ B \subset C \iff \forall x (x\in B \implies x \in C) $$ By the syllogism, $$ \forall x (x\in A \implies x \in C) \iff A \subset C $$

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