subset

Set relations

https://en.wikipedia.org/wiki/Subset

Two sets, A and B, may be related (or interact) in numerous ways:

• Disjoint sets: A and B do not interact, they're not related in any way; they are completely disjoint (non-overlapping) from one another.

Sets $x$ and $y$ may interact or be related in several ways:

• $x$ is a member of $y$: $x\in y$ (membership relation)

• $x$ is a subset of $y$: $x\subseteq y$ (inclusion relation)

  • $x$ is a proper subset of $y$: $x\subset y$

  • $x$ is equal to $y$ if $x\subseteq y \land y\subseteq x$

Axioms:

• every set is a subset of itself, $A\subseteq A$

• the empty set is a subset of every set, $\varnothing \subseteq A$

• therefore, the empty set is a subset of itself, $\varnothing \subseteq \varnothing$

Subset relation

Besides membership (which relates elements and sets), another fundamental relation is the subset relation, also called the inclusion relation, which is relation between sets, denoted by the symbol $\subseteq$.

If all elements of a set $X$ are also elements of a set $Y$, then $X$ is the subset of $Y$, denoted by

$X \subseteq Y$

At the same time, set $Y$ is a superset of $X$, denoted by $Y \supseteq X$

Given objects x and y
(y = {x}) → (∀x . x ∈ y ∧ [x ⊆ y → ∀z.z ∈ x ∧ z ∈ y])
  then x ∈ y holds always,
  but  x ⊆ y holds only if x is a set, e.g.:
    (x=1 → y={1}) → (x ∈ y ∧ x ⊈ y)
    if (x={1}) → (y={{1}}), then x ∈ y and x ⊆ y

• ∈ ∋ ∉ ∌

• ∣ ∅ ⋜ ⋝

• ∃ ∀ ∄

• ⋃ ⋂

• ⊆ ⊇ ⊂ ⊃

• ⊊ ⊋ ⊄ ⊅

• ⊈ ⊉

$$\text{Given } a,b\ where\ b = \{a\} \\ \text{if } a = 1\ \text{then } b = \{1\}\ ,so\ a \in b\\ \text{if } a = \{1\}\ \text{then } b = \{\{1\}\}\ ,so\ a\in b \land a \subseteq b\\$$

A set $X$ is a proper subset of a set $Y$, denoted by $X\subset Y$ if set $Y$ has additional elements besides those that are also in set $X$.

That is, if every element of $X$ is an element of $Y$ and $|X| < |Y|$.

For example, $X=\{1,2\}$ and $Y=\{1,2,3,4\}$.

Here, $X\subseteq Y$, but, more precisely, $X$ is a proper subset of $Y$ i.e. $X\subset Y$.

Every set is a subset of itself: $\{a,b,c\} \subseteq \{a,b,c\}$