DS101 MIDTERM REVIEWER

Module 5-8

Conditional Statements

are the foundation of logical reasoning and computer programming.

Converse

formed by reversing the hypothesis and conclusion.

If q, then p.

Converse Statement Form

Inverse

formed by negating both the hypothesis and conclusion.

If not p, then not q.

Inverse Statement Form

Contrapositive

formed by reversing and negating both the hypothesis and conclusion.

If not q, then not p.

Contrapositive Statement Form

Generalization

It allows us to make a broader statement from something that is already true. If one statement is true, then we can also say that “it or something else” is true.

p ∴ p v q

q ∴ p v q

Generalization Symbol Form

Specialization

It allows us to take a specific part of a statement that contains two or more facts.

p ^ q ∴ p

p ^ q ∴ q

Specialization Symbol Form

Elimination

It says that if you have two possible statements and one is false, then the other must be true.

p v q, ~q ∴ p

p v q, ~p ∴ q

Elimination Symbol Form

Transitivity

It states that if one statement leads to a second, and the second leads to a third, then the first also leads directly to the third.

p ⟶ q

q ⟶ r

∴ p ⟶ r

Transitivity Symbol Form

Proof by Division into Cases

It says that if there are two possible situations, and both lead to the same conclusion, then that conclusion must be true.

p v q,

p ⟶ r,

q ⟶ r

∴ r

Proof by Division into Cases Symbol Form

Fallacy

a mistake in reasoning. It looks logical, but it’s actually wrong.

• Avoid wrong conclusions

• Strengthen our reasoning

• Spot false arguments in debates or media

Why do we need to study fallacies?

Formal Fallacy

error in the structure of logic.

Informal Fallacy

error in the content or meaning of words.

Ad Hominem

False Cause

Hasty Generalization

Appeal to Authority

Bandwagon

Examples of Informal Fallacy

Ad Hominem

Attacking the person, not the idea.

Example: Don’t listen to Maria’s opinion on climate change; she failed her science class.

False Cause

Assuming one thing caused another.

Example: I wore my lucky shirt during the exam and got a high score, so the shirt brought me good luck.

Hasty Generalization

Quick judgment from few examples.

Example: Two students from that school were rude, so everyone there must be rude.

Appeal to Authority

It’s true because an expert said so.

Example: This energy drink must be healthy because a famous doctor on TV recommends it.

Bandwagon

Everyone’s doing it, so it’s right.

Example: All my friends are skipping class, so it must be okay to skip too.

Fallacy of Language

happens when words or grammar cause confusion or false meaning.

Language shapes how we think and reason.

Why Language Matters in Logic

Ambiguity

Amphiboly

Equivocation

Composition

Division

Accent

Types of Fallacy of Language

Ambiguity

a word has multiple meanings.

Ex.

“I saw her duck.”⟶ (It’s unclear whether ‘duck’ means the animal or the action of lowering her head.)

Amphiboly

unclear grammar or structure.

Ex. “The teacher told the student that he was lazy. ” ⟶ (It’s unclear who ‘he’ refers to — the teacher or the student.)

Equivocation

one word used in two meanings.

Ex. “A feather is light. What is light cannot be dark. Therefore, a feather cannot be dark. ” ⟶ (The word “light” changes meaning from ‘not heavy’ to ‘not dark’ )

Composition

thinking what’s true for parts is true for the whole.

Ex. “Each player is good, so the team must be good. ” ⟶ (What’s true of the parts may not be true of the whole.)

Division

opposite of composition.

Ex. “Our school is famous, so every student here must be famous too. ” ⟶ (What’s true of the whole doesn’t apply to every part.)

Accent

changing meaning by stress or emphasis.

Ex. “I didn’t steal his money. ” ⟶ (Depending on which word is emphasized, the meaning changes, maybe someone else did, or it wasn’t money.)

Principle of Counting

If one event can happen in m ways and another in n ways, then both can happen in m × n ways.

1. Multiplication Rule (AND)

2. Addition Rule (OR)

2 Basic Rules of Counting

Multiplication Rule (AND)

When events happen together, multiply.

Example: 3 shirts × 2 pants = 6 outfits

Addition Rule (OR)

When one of several can happen, add.

Example: 5 math OR 4 computer subjects → 5 + 4 = 9 choices

Factorial

counts arrangements of items.

Pascal’s Triangle

• A triangle of numbers that shows patterns in counting.

• Each number = sum of two numbers above it.

• Used to find combinations and binomial coefficients.

nCr = n! / [r!(n − r)!]

Pascal’s Triangle Formula

⟶ Password creation

⟶ Arranging data

⟶ Computing probabilities

Real-Life Applications: Counting Principles

⟶ Probability & statistics

⟶ Pattern recognition

⟶ Network path counting

Real-Life Applications: Pascal’s Triangle