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For some set \(\mathbb A\) the image of \(\mathbb A\) by \(f\) is \(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\). We can also define the inverse of an image even when the function itself isn’t invertible: \(f^{-1}(\mathbb A)\)
+
+Combinatorics
+
+- \(|\mathbb A|\) size (number of elements) of \(\mathbb A\)
+- Rule of addition:
+
+- If \(\mathbb A, \mathbb B \subseteq \mathbb X\) and \(|\mathbb A|, |\mathbb B| \in \mathbb N\) and \(\mathbb A \cap \mathbb B = \emptyset\) then \(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)
+- Can be generalized as: \[
+(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
+|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
+(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
+\]
+
+- Rule of multiplication:
+
+- \(\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|\)
+- Can be generalized as: \[
+(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
+|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
+\]
+