Finished ex1 from dstr hw2

1 files changed, 4 insertions, 3 deletions

diff --git a/Year 1/Discrete math/Homeworks/Homework2.tex b/Year 1/Discrete math/Homeworks/Homework2.tex
index 4eaaf7e..54c19d4 100644
--- a/Year 1/Discrete math/Homeworks/Homework2.tex
+++ b/Year 1/Discrete math/Homeworks/Homework2.tex
@@ -42,9 +42,10 @@
 \begin{align*}
 	& \textbf{База: } n = 1, \quad S(1) = \sum_{X \subseteq \{ 1 \}} \prod_{y \in X} y = \prod_{y \in \emptyset} y + \prod_{y \in {1}} y = 1 + 1 = 2 \quad (1+1)! = 2! = 2 \\
 	& \textbf{Индукционно предположение: } S(n) = (n+1)! \text{ е изпълнено за поне едно } n \in \mathbb{N}^+ \\
-	& \textbf{Индукционна стъпка: } S(n+1) = \sum_{X \subseteq \{ 1, ..., n+1 \}} \prod_{y \in X} y = \sum_{X \subseteq \{ 1, ..., n \}} \prod_{y \in X} y + (n+1) \prod_{y \in \{1, ..., n+1\}} y \xlongequal{\text{ИП}} \\
-	& \qquad \xlongequal{\text{ИП}} (n+1)! + (n+1) \prod_{y \in \{1, ..., n+1\}} y = (n+1)! + (n+1) \cdot 1 \cdot ... \cdot (n+1) = (n+1)! + (n+1)(n+1)! = \\
-	& \qquad = (n+1)!(n+1+1) = (n+2)!
+	& \textbf{Индукционна стъпка: } S(n+1) = \sum_{X \subseteq \{ 1, ..., n+1 \}} \prod_{y \in X} y = \sum_{X \subseteq \{ 1, ..., n \}} \prod_{y \in X} y \; + \\
+	& + \prod_{y \in \{ n+1 \}} y + \prod_{y \in \{ 1, n+1 \}} y + \prod_{y \in \{ 2, n+1 \}} y + \cdots + \prod_{y \in \{ n, n+1 \}} y + \prod_{y \in \{ 1, 2, n+1 \}} y + \cdots + \prod_{y \in \{ 1, ..., n, n+1 \}} y = \\
+	& = \sum_{X \subseteq \{ 1, ..., n \}} \prod_{y \in X} y + (n+1) \left( \prod_{y \in \emptyset} y + \prod_{y \in \{ 1 \}} y + \prod_{y \in \{ 2 \}} y + \cdots + \prod_{y \in \{ n \}} y + \prod_{y \in \{ 1, 2 \}} y + \cdots + \prod_{y \in \{ 1, ..., n \}} y \right) = \\
+	& = \sum_{X \subseteq \{ 1, ..., n \}} \prod_{y \in X} y + (n+1)\sum_{X \subseteq \{ 1, ..., n \}} \prod_{y \in X} y \xlongequal{\text{ИП}} (n+1)! + (n+1)(n+1)! = (n+1)!(n+1+1) = (n+2)!
 \end{align*}
 
 \paragraph{\colorbox{yellow}{Задача 2} Решение:} [TODO]