Welcome to Week 1 of MAT135: Topics in Contemporary Mathematics! This week, we aren't just crunching numbers; we are learning how to think critically about the world around us. The focus of this week's notes covers Chapter 1: Critical Thinking, moving from public policy data to the formal logic used by computers.

Section 1.1: Public Policy and Simpson's Paradox

Have you ever heard the phrase "numbers don't lie"? While the math might be correct, the way data is grouped can be incredibly deceptive. This week we studied Simpson's Paradox, a phenomenon where a trend appears in several different groups of data but disappears or reverses when these groups are combined.

We looked at examples ranging from Berkeley graduate admissions to baseball batting averages. The key takeaway is that an "average" is not always a fair representation of performance if the underlying variables (like difficulty of department or number of at-bats) aren't weighted correctly.

Quick Math Review: Remember the formula for calculating percentages, which is crucial for comparing these datasets:

$$ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% $$

Section 1.2: Logic and Informal Fallacies

Logic is the study of methods used to distinguish correct reasoning from incorrect reasoning. We discussed two types of arguments:

  • Deductive Arguments: Drawing a conclusion from premises based on strict logic (e.g., All men are mortal $\rightarrow$ Socrates is a man $\rightarrow$ Socrates is mortal).
  • Inductive Arguments: Drawing a general conclusion from specific examples/patterns.

We also explored Informal Fallacies—arguments that seem correct on the surface but are logically flawed. Some common ones to watch out for include:

  • The Straw Man: Dismissing a distorted version of an opponent's argument rather than the actual argument.
  • False Cause: Assuming that because two events occur together, one caused the other.
  • False Authority: Accepting a claim based on an authority figure whose expertise is irrelevant to the topic (e.g., an actor giving medical advice).

Section 1.3: Formal Logic and Truth Tables

Finally, we looked at how computers "think" using formal logic. We use variables like $p$ and $q$ to represent statements and analyze them using Truth Tables.

Key logical operations include:

  • Negation (NOT $p$): The opposite of a statement.
  • Conjunction ($p$ AND $q$): True only if both statements are true.
  • Disjunction ($p$ OR $q$): True if at least one statement is true.
  • Conditional ($p \rightarrow q$): The "If... then" statement.

A major focus was understanding the variations of the conditional statement ($p \rightarrow q$):

  • Converse: $q \rightarrow p$
  • Inverse: $(\text{NOT } p) \rightarrow (\text{NOT } q)$
  • Contrapositive: $(\text{NOT } q) \rightarrow (\text{NOT } p)$

Note: A conditional statement is logically equivalent to its contrapositive!

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