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tanszek:oktatas:techcomm:information_-_basics:sciences

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tanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/08 19:18] – kneheztanszek:oktatas:techcomm:information_-_basics:sciences [2025/09/08 19:42] (current) – [Deductive Sciences] knehez
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 According to **principles**, **conditions** (circumstances), and **results** (these three general aspects) we can categorize every scientific problem into the following problem groups. According to **principles**, **conditions** (circumstances), and **results** (these three general aspects) we can categorize every scientific problem into the following problem groups.
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 +----
  
 ====== Inductive Sciences ====== ====== Inductive Sciences ======
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 **Remark**: Legislative processes are based on an inductive method that analyzes social problems and their causes and makes new laws as a conclusion. **Remark**: Legislative processes are based on an inductive method that analyzes social problems and their causes and makes new laws as a conclusion.
  
-**Example**: In **information technology**, mathematical induction can be applied to many areas, including algorithm analysis and data structures. One typical example is proving the correctness of algorithms or the properties of data structures like trees. Here’s an example from **binary trees**:+---- 
 + 
 +**Example 1**: In **information technology**, mathematical induction can be applied to many areas, including algorithm analysis and data structures. One typical example is proving the correctness of algorithms or the properties of data structures like trees. Here’s an example from **binary trees**:
  
 **Problem**: **Problem**:
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   - **Inductive Hypothesis**: Assume that for any binary tree with \(k\) nodes, the number of edges is \(k-1\).   - **Inductive Hypothesis**: Assume that for any binary tree with \(k\) nodes, the number of edges is \(k-1\).
   - **Inductive Step**: We must prove that if the statement holds for a binary tree with \(k\) nodes, then it also holds for a binary tree with \(k+1\) nodes. \\ Suppose we add one more node to the binary tree, bringing the total number of nodes to \(k+1\). When we add this node, we also add exactly one edge connecting the new node to an existing node in the tree (either as a left or right child of a parent node). \\ \\ By the inductive hypothesis, the tree with \(k\) nodes has \((k - 1)\) edges. Adding one more node introduces one additional edge, so the number of edges in the tree with \((k + 1)\) nodes is: $$ (k-1) + 1 = k $$ This matches the formula for the number of edges in a tree with \((k + 1)\) nodes, which should be \((k-1) + 1 = k\).    - **Inductive Step**: We must prove that if the statement holds for a binary tree with \(k\) nodes, then it also holds for a binary tree with \(k+1\) nodes. \\ Suppose we add one more node to the binary tree, bringing the total number of nodes to \(k+1\). When we add this node, we also add exactly one edge connecting the new node to an existing node in the tree (either as a left or right child of a parent node). \\ \\ By the inductive hypothesis, the tree with \(k\) nodes has \((k - 1)\) edges. Adding one more node introduces one additional edge, so the number of edges in the tree with \((k + 1)\) nodes is: $$ (k-1) + 1 = k $$ This matches the formula for the number of edges in a tree with \((k + 1)\) nodes, which should be \((k-1) + 1 = k\). 
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 +----
 +
 +**Example 2**: Sum of consecutive natural numbers. **Claim**:
 +
 +$$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. $$
 +
 +Base case (n=1):
 +
 +$$ 1 = \frac{1 \cdot 2}{2} = 1, $$
 +
 +so the statement holds.
 +
 +**Inductive step**:
 +
 +Assume the formula is true for \(n = k\):
 +
 +$$ 1 + 2 + \cdots + k = \frac{k(k+1)}{2}. $$
 +
 +**Simplify**:
 +$$ \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}. $$
 +
 +Thus, the formula also holds for \( n = k+1 \).
  
 https://en.wikipedia.org/wiki/Mathematical_induction https://en.wikipedia.org/wiki/Mathematical_induction
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 +----
  
 ====== Deductive Sciences ====== ====== Deductive Sciences ======
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 The quote from [[https://en.wikipedia.org/wiki/Niels_Bohr|Niels Bohr]], //"In the Institute, we only have one piece of experimental equipment: a ping-pong table"// is a good example of reductive reasoning in science. It suggests that groundbreaking discoveries can be achieved not through extensive experimental setups but rather through the adjustment of theoretical laws and conditions. Bohr highlights the power of thought experiments and abstract reasoning, emphasising that manipulating underlying principles can lead to new insights without always needing physical experimentation. The quote from [[https://en.wikipedia.org/wiki/Niels_Bohr|Niels Bohr]], //"In the Institute, we only have one piece of experimental equipment: a ping-pong table"// is a good example of reductive reasoning in science. It suggests that groundbreaking discoveries can be achieved not through extensive experimental setups but rather through the adjustment of theoretical laws and conditions. Bohr highlights the power of thought experiments and abstract reasoning, emphasising that manipulating underlying principles can lead to new insights without always needing physical experimentation.
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 +----
  
 ====== Reductive Sciences ====== ====== Reductive Sciences ======
tanszek/oktatas/techcomm/information_-_basics/sciences.1757359080.txt.gz · Last modified: 2025/09/08 19:18 by knehez