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Signed-off-by: Abdulkadir Furkan Şanlı <abdulocracy@disroot.org>
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Abdulkadir Furkan Şanlı 2019-11-27 15:38:07 +01:00
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</ul></li>
</ul></li>
<li>For some set <span class="math inline">\(\mathbb A\)</span> the image of <span class="math inline">\(\mathbb A\)</span> by <span class="math inline">\(f\)</span> is <span class="math inline">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn’t invertible: <span class="math inline">\(f^{-1}(\mathbb A)\)</span></li>
</ul>
<h2 id="combinatorics">Combinatorics</h2>
<ul>
<li><span class="math inline">\(|\mathbb A|\)</span> size (number of elements) of <span class="math inline">\(\mathbb A\)</span></li>
<li>Rule of addition:
<ul>
<li>If <span class="math inline">\(\mathbb A, \mathbb B \subseteq \mathbb X\)</span> and <span class="math inline">\(|\mathbb A|, |\mathbb B| \in \mathbb N\)</span> and <span class="math inline">\(\mathbb A \cap \mathbb B = \emptyset\)</span> then <span class="math inline">\(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)</span></li>
<li>Can be generalized as: <span class="math display">\[
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
\]</span></li>
</ul></li>
<li>Rule of multiplication:
<ul>
<li><span class="math inline">\(\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|\)</span></li>
<li>Can be generalized as: <span class="math display">\[
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
\]</span></li>
</ul></li>
</ul>
</div>