# Closure for Finite Collection of Relations and Operations

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## Theorem

Let $\RR_1, \RR_2, \ldots \RR_n$ be relations.

Let $\SS_1, \SS_2, \ldots \SS_m$ be operations.

Let $T$ be a small class.

Let the image of $\RR_i$ over any small class $x$ be small classes for $1 \le i \le n$.

Let the image of $\SS_i$ over any Cartesian product $x \times x$ be small classes for $1 \le i \le m$.

Then there exists a small class $X$ such that:

- $(1): \quad$ $T \subseteq X$
- $(2): \quad$ Each $\RR_i$ is closed with respect to $X$. Each $\SS_i$ is closed with respect to $X \times X$.
- $(3): \quad$ $X$ is the smallest small class satisfying conditions $(1)$ and $(2)$ above.

If $Y$ satisfies conditions $(1)$ and $(2)$, then $X \subseteq Y$.

## Proof

Let $R \sqbrk x$ denote the image of $x$ under $R$.

Set:

- $\ds \map G x = x \cup \bigcup_{i \mathop = 1}^n \RR_i \sqbrk x \cup \bigcup_{i \mathop = 1}^m \SS_i \sqbrk x$

Using the Principle of Recursive Definition, construct the function $F$ as follows:

- $\map F 0 = T$
- $\map F {n + 1} = \map G {\map F n}$

Define $X$ as follows:

- $\ds X = \bigcup_{n \mathop \in \omega} \map F n$

### Proof that $T \subseteq X$

\(\ds T\) | \(=\) | \(\ds \map F 0\) | Definition of $F$ | |||||||||||

\(\ds \) | \(\subseteq\) | \(\ds \bigcup_{n \mathop \in \omega} \map F n\) | Set is Subset of Union |

$\Box$

### Proof of Closure

Take any $\RR_i$.

Suppose $x \mathrel {\RR_i} y$ and $x \in \map F n$ for some $n$.

Then $y \in \map F {n + 1}$ by definition of image.

Thus $y \in X$ and $R_i$ is closed with respect to $X$.

Similarly, take any $\SS_i$.

Suppose $\paren {x \mathrel {S_i} y} = z$ and that $x \in X$ and $y \in X$.

It follows that $x, y \in \map F n$ for some $n$.

So:

- $z \in \map F {n + 1}$

Therefore:

- $z \in X$

$\Box$

### Proof that $X$ is the Smallest Relational Closure

Suppose $T \subseteq Y$ and that $R_i$ and $S_i$ are closed with respect to $Y$.

$\map F n \subseteq Y$ shall be proven by induction:

For all $n \in \omega$, let $\map P n$ be the proposition:

- $\map F n \subseteq Y$

### Basis for the Induction

$\map P 0$ is the case:

- $\map F 0 \subseteq Y$

which has been proved above.

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

- $\map F k \subseteq Y$

Then we need to show:

- $\map F {k + 1} \subseteq Y$

### Induction Step

This is our induction step:

\(\ds \map F k\) | \(\subseteq\) | \(\ds Y\) | Inductive Hypothesis | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds R_i \sqbrk {\map F k}\) | \(\subseteq\) | \(\ds Y\) | |||||||||||

\(\, \ds \land \, \) | \(\ds S_i \sqbrk {\map F k \times \map F k}\) | \(\subseteq\) | \(\ds Y\) | Image Preserves Subsets and $R_i$ and $S_i$ are closed with respect to $Y$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map F k \cup \bigcup_{i \mathop = 1}^n R_i \sqbrk {\map F k} \cup \bigcup_{i \mathop = 1}^m S_i \sqbrk {\map F k}\) | \(\subseteq\) | \(\ds Y\) | Indexed Union Subset | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map G {\map F k}\) | \(\subseteq\) | \(\ds Y\) | Definition of $G$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map F {k + 1}\) | \(\subseteq\) | \(\ds Y\) | Definition of $\map F {k + 1}$ |

So $\map P k \implies \map P {k + 1}$ and the result follows by Principle of Mathematical Induction.

Therefore:

- $\forall n \in \omega: \map F n \subseteq R$

Hence by Indexed Union Subset:

- $T \subseteq Y$

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 9.6$