Induction, Deduction and Reduction

归纳与演绎解析

1. Description 1

induction是指归纳,将具体的归纳出一个结论,而这个结论不一定就是正确的。而deduction演绎是指根据一个已知的定论描述的更具体,所以某种程度,它们两是OPPOSITE

2. Description 2

 induction:归纳,指由特殊的事实或例子推理出的普遍性规律的过程,是证明涉及到正整数变量的定理所运用的两部法。第一是证明定理对于最小整数值成立,第二是证明若定理对于任何整数值都成立,则它也必然对下一个整数值成立。
  deduction:演绎,指推理的过程,在此过程中必然可从所述前提得出一个结论,即从一般推向特殊的推论。
  reduction:约分或简化,指削去分数中分子和分母公约数、把分数写成与之相等的小数或把表达式或方程式化至最简的形式。

3. Description 3

Induction为归纳(从特例到普遍),Deduction is 推导(从普遍到特例),Reduction is 约分.

4. Description 4

归纳和推演:Induction vs. Deduction
http://blog.sina.com.cn/s/blog_4dd6d4750102e5mt.html

Weak induction and strong induction are both proof techniques used in mathematics, particularly in the context of proving statements about natural numbers. ### Weak Induction Weak induction, also known as simple induction, follows a two - step process: 1. **Base Case**: Prove that the statement is true for the smallest value of \(n\) (usually \(n = 0\) or \(n=1\)). 2. **Inductive Step**: Assume that the statement is true for some arbitrary natural number \(k\) (the inductive hypothesis). Then, prove that the statement is true for \(k + 1\). The general form of a weak induction proof can be presented in a Python - like pseudocode for the concept: ```python def weak_induction_proof(n): # Base case if n == base_case_value: return prove_base_case() # Inductive step inductive_hypothesis = weak_induction_proof(n - 1) if inductive_hypothesis: return prove_inductive_step(n) return False ``` ### Strong Induction Strong induction is a variation of weak induction. Instead of just assuming that the statement is true for a single value \(k\), in strong induction, the inductive hypothesis assumes that the statement is true for all values from the base case up to an arbitrary natural number \(k\). 1. **Base Case**: Similar to weak induction, prove that the statement is true for the smallest value of \(n\). 2. **Inductive Step**: Assume that the statement is true for all natural numbers \(m\) where \(m\) ranges from the base case value up to \(k\). Then, prove that the statement is true for \(k + 1\). The general form of a strong induction proof in a Python - like pseudocode for the concept: ```python def strong_induction_proof(n): # Base case if n == base_case_value: return prove_base_case() # Inductive step for m in range(base_case_value, n): if not strong_induction_proof(m): return False return prove_inductive_step(n) ``` In summary, the main difference between weak and strong induction lies in the inductive hypothesis. Weak induction only assumes the truth of the statement for one previous value, while strong induction assumes the truth for all previous values up to the given value.
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