The Clowder Project

Let $X$ be a set.

1. 1.

   Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors

   \[ \begin{array}{ccc} U\setminus -\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\supset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\setminus V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\setminus -_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]

   In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:

   1. (a)

      If $U\subset A$, then $U\setminus V\subset A\setminus V$.

   2. (b)

      If $V\subset B$, then $U\setminus B\subset U\setminus V$.

   3. (c)

      If $U\subset A$ and $V\subset B$, then $U\setminus B\subset A\setminus V$.

2. 2.

   De Morgan’s Laws. We have equalities of sets

   \begin{align*} X\setminus \webleft (U\cup V\webright ) & = \webleft (X\setminus U\webright )\cap \webleft (X\setminus V\webright ),\\ X\setminus \webleft (U\cap V\webright ) & = \webleft (X\setminus U\webright )\cup \webleft (X\setminus V\webright ) \end{align*}

   for each $U,V\in \mathcal{P}\webleft (X\webright )$.

3. 3.

   Interaction With Unions I. We have equalities of sets

   \[ U\setminus \webleft (V\cup W\webright )=\webleft (U\setminus V\webright )\cap \webleft (U\setminus W\webright ) \]

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

4. 4.

   Interaction With Unions II. We have equalities of sets

   \[ \webleft (U\setminus V\webright )\cup W=\webleft (U\cup W\webright )\setminus \webleft (V\setminus W\webright ) \]

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

5. 5.

   Interaction With Unions III. We have equalities of sets

   \begin{align*} U\setminus \webleft (V\cup W\webright ) & = \webleft (U\cup W\webright )\setminus \webleft (V\cup W\webright )\\ & = \webleft (U\setminus V\webright )\setminus W\\ & = \webleft (U\setminus W\webright )\setminus V \end{align*}

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

6. 6.

   Interaction With Unions IV. We have equalities of sets

   \[ \webleft (U\cup V\webright )\setminus W=\webleft (U\setminus W\webright )\cup \webleft (V\setminus W\webright ) \]

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

7. 7.

   Interaction With Intersections. We have equalities of sets

   \begin{align*} \webleft (U\setminus V\webright )\cap W & = \webleft (U\cap W\webright )\setminus V\\ & = U\cap \webleft (W\setminus V\webright ) \end{align*}

   for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

8. 8.

   Interaction With Complements. We have an equality of sets

   \[ U\setminus V=U\cap V^{\textsf{c}} \]

   for each $U,V\in \mathcal{P}\webleft (X\webright )$.

9. 9.

   Interaction With Symmetric Differences. We have an equality of sets

   \[ U\setminus V=U\mathbin {\triangle }\webleft (U\cap V\webright ) \]

   for each $U,V\in \mathcal{P}\webleft (X\webright )$.

10. 10.

    Triple Differences. We have

    \[ U\setminus \webleft (V\setminus W\webright )=\webleft (U\cap W\webright )\cup \webleft (U\setminus V\webright ) \]

    for each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

11. 11.

    Left Annihilation. We have

    \[ \text{Ø}\setminus U=\text{Ø} \]

    for each $U\in \mathcal{P}\webleft (X\webright )$.

12. 12.

    Right Unitality. We have

    \[ U\setminus \text{Ø}=U \]

    for each $U\in \mathcal{P}\webleft (X\webright )$.

13. 13.

    Right Annihilation. We have

    \[ U\setminus X=U \]

    for each $U\in \mathcal{P}\webleft (X\webright )$.

14. 14.

    Invertibility. We have

    \[ U\setminus U = \text{Ø} \]

    for each $U\in \mathcal{P}\webleft (X\webright )$.

15. 15.

    Interaction With Containment. The following conditions are equivalent:

    1. (a)

       We have $V\setminus U\subset W$.

    2. (b)

       We have $V\setminus W\subset U$.

16. 16.

    Interaction With Characteristic Functions. We have

    \[ \chi _{U\setminus V}=\chi _{U}-\chi _{U\cap V} \]

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

17. 17.

    Interaction With Direct Images. We have a natural transformation

    
    with components
    \[ f_{!}\webleft (U\webright )\setminus f_{!}\webleft (V\webright )\subset f_{!}\webleft (U\setminus V\webright ) \]

    indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.

18. 18.

    Interaction With Inverse Images. The diagram

    
    commutes, i.e. we have
    \[ f^{-1}\webleft (U\setminus V\webright )=f^{-1}\webleft (U\webright )\setminus f^{-1}\webleft (V\webright ) \]

    for each $U,V\in \mathcal{P}\webleft (X\webright )$.

19. 19.

    Interaction With Codirect Images. We have a natural transformation

    
    with components
    \[ f_{!}\webleft (U\webright )\setminus f_{!}\webleft (V\webright )\subset f_{!}\webleft (U\setminus V\webright ) \]

    indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.