Logic, Two-Way Tables, and Problem Solving Review

Course Logistics & Policies

• Exam Note Card: A $3 \times 5\text{ inch}$ note card (front and back), a pen or pencil, scrap paper, and a calculator (graphing calculator recommended) are permitted.
• Exam Format: The exam contains $15$ to $20$ questions.
• Attendance Threshold: Missing $20\%$ of the class sessions ($3$ classes) results in course withdrawal.

Inductive vs. Deductive Reasoning

• Inductive Reasoning:
  • Arguments progress from specific premises to a general conclusion.
  • Evaluated as strong (sufficient supporting evidence) or weak (insufficient evidence); never evaluated as true/false or valid/invalid.
• Deductive Reasoning:
  • Arguments progress from general premises to a specific conclusion.
  • Evaluated for validity (whether the conclusion logically follows from the premises).
  • Valid arguments are further evaluated for soundness:
  • Sound: Valid argument with factually true premises.
  • Unsound: Valid argument with one or more false premises.
  • Invalid arguments cannot be sound.
• Logical Rules & Fallacies:
  • Valid argument form: If $p \implies q$ and $p$ is true, then $q$ is true.
  • Invalid argument fallacy: If $p \implies q$ and $q$ is true, it does not guarantee $p$ is true.
• Venn Diagrams:
  • Used to diagram set inclusion and verify deductive validity. An argument is valid if the element $x$ lands in the target set indicated by the conclusion.

Two-Way Tables & Percentages

• Percentage Calculations:
  • Convert percentages to decimals by moving the decimal point two places left.
  • Replace "of" with multiplication ($\times$).
• Handedness Distribution Sample:
  • Men ($350$ total, $14\%$ left-handed):
  • Left-handed: $0.14 \times 350 = 49$
  • Right-handed: $350 - 49 = 301$
  • Women ($100$ total, $8\%$ left-handed):
  • Left-handed: $0.08 \times 100 = 8$
  • Right-handed: $100 - 8 = 92$

Problem Solving Process & Unit Analysis

• Three-Step Problem Solving Process:
  1. Understand the Problem: Determine target quantities, draw diagrams, and formulate solution plans.
  2. Solve the Problem: Gather data, maintain organized steps, and verify calculations.
  3. Explain Results: Verify answer reasonableness, attach appropriate units, and state findings clearly.
• Unit Analysis Rules:
  • Quantities with different units cannot be added or subtracted.
  • Quantities with different units can be multiplied or divided.
  • Operational keywords:
  • "Per" indicates division ($/$).
  • "Of" or hyphen indicates multiplication ($\times$).
  • "Square" indicates second power ($\text{unit}^2$), as in $\text{Area} = \text{length} \times \text{width}$.
  • "Cube" or "Cubic" indicates third power ($\text{unit}^3$), as in $\text{Volume} = \text{length} \times \text{width} \times \text{height}$.