Section 1.1: Inductive and Deductive Reasoning (Notes)

Inductive Reasoning

• Overview: Arriving at a general conclusion based on observations of specific examples, looking for patterns to predict future behavior. Closely related to the scientific method (observing, forming hypotheses/conjectures).
• Caveat: Inductive conclusions (conjectures) may not be true; a single counterexample can disprove them. Induction does not yield proven truths.
• Key terms:
  • Conjecture / Hypothesis: A conclusion from inductive reasoning, may or may not be true.
  • Counterexample: A case where the conjecture fails, used to prove inductive claims false.
• Strong vs. weak inductive arguments:
  • Strong: Large, random sample size increases reliability (e.g., $n = 380{,}000$ freshmen, $25\%$ unprepared, concluding $0.2484 \le p \le 0.2525$).
  • Weak: Small or non-random sample, leading to bias (e.g., concluding all men have difficulty expressing feelings based on two personal experiences).
• In mathematics (patterns and predictions):
  • Example 1 (Arithmetic): Sequence $3, ext{ }12, ext{ }21, ext{ }30, ext{ }39$. Common difference is $9$, next term is $48$.
  • Example 2 (Multiplicative): Sequence $3, ext{ }12, ext{ }48, ext{ }192, ext{ }768$. Each term is four times the previous, next term is $3072$.
  • Example 3 (Fibonacci): Sequence $1, ext{ }1, ext{ }2, ext{ }3, ext{ }5, ext{ }8, ext{ }13, ext{ }21, ext{ }34$. Each term is the sum of the two previous terms ($F<em>n = F</em>{n-1} + F_{n-2}$).
  • Example 4 (Concurrent patterns): Shapes alternating circle/square while magenta dots rotate counterclockwise.
• Downsides: Not guaranteed true, disprovable by counterexample, quality depends on sample representativeness and avoiding bias.

Deductive Reasoning

• Overview: Proving a specific conclusion from general statements, axioms, definitions, or theorems. Aims to establish truth if the general statements are true.
• Distinction from Induction:
  • Induction: Observation-based, generates hypotheses, not a proof.
  • Deduction: Starts from established general truths, derives specific conclusions, aims at proofs/theorems.
• Structure in mathematics: Starts from accepted true general statements (axioms, lemmas, theorems), uses logical steps to derive a conclusion (a theorem if valid).
• Everyday example (Scrabble):
  • Premises: (1) Scrabble prohibits all proper names; (2) Texas is a proper name.
  • Conclusion: Therefore, the word Texas is prohibited in Scrabble.
• Mathematical example (Algorithm):
  • Problem: Apply steps to a number (multiply by 6, add 8, divide by 2, subtract 4).
  • Deductive approach (algebraic): Apply to input $n$:
  1. $6n$
  2. $6n + 8$
  3. $\frac{6n + 8}{2} = 3n + 4$
  4. $(3n + 4) - 4 = 3n$
  • Conclusion: The entire sequence of steps is equivalent to multiplying the original number by $3$ ($$