Set theory

Summary:

Name	Notation ( $\LaTeX$ )
Union	$A \cup B$ (cup)
Intersection	$A \cap B$ (cap)
Set difference	$A \backslash B$ (backslash)
Symmetric difference	$A \triangle B$ (triangle)
Cartesian product	$A \times B$ (times)
Power set	$\mathcal{P}(A)$ (mathcal{P})

Union

For sets $A,B$ , the union is written as $A \cup B$ , which is the set of all objects that are members of $A$ or $B$ or both.

$A=\{1,2,3\}, B=\{2,3,4\}, A\cup B = \{1,2,3,4\}$

Intersection

For sets $A,B,$ the intersection is written as $A \cap B$ , and is the set of objects contained in both $A$ and $B$ :

$A=\{1,2,3\}, B=\{2,3,4\}, A\cup B = \{2,3\}$

Difference

Set difference

The asymmetric difference between two sets, for example $A \backslash B$ , is the subset of $A$ that is not in $B$ , and for $B \backslash A$ the converse:

$A \backslash B = \{1\}, B \backslash A = \{4\}$

Symmetric difference

The symmetric difference of $A,B$ is written as $A \triangle B$ or $A \ominus B$ , and comprises the mutually exclusive members of $A,B$ :

$A \triangle B = \{1,4\}$

The symmetric difference is also equivalent to the set difference of the union and the intersection:

$$ (A \cup B) \backslash (A \cap B)$$

Cartesian product

The Cartesian product of two sets $A,B$ gives all possible ordered pairs $(a,b)$ :

$A \times B = \{1,2\}, \{1,3\}, \ldots \{3,4\}$

Power set

The power set of $A$ , written as $\mathcal{P}(A)$ , comprises all subsets of $A$ :

\mathcal{P}(A) = \{ \{\}, \{1\}, \{2\}, \{1,2\}, \ldots \} export const _frontmatter = {}

Backlinks

Shapley values

To break this down intuitively: the summation occurs over every possible coalition (i.e. combinations of players) excluding a single player (the set difference $N \backslash \{i\}$, see [[set-theory]]). The last term in the equation equates to the difference in value with [$v(S \cup \{i\})$] and without [$v(S)$] player $i$.[^1] The Shapley value given by the *average* contribution of a player computed this way.