Complete Logic & Proof Study Guide

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🧠 Complete Logic & Proof Study Guide

Based on your folder materials:

Writing Proofs,

Deductive Reasoning,

Statements, Conditionals, Biconditionals, and

Conjectures and Counterexamples.

1. Logic Foundations

Basic Definitions

  • Statement: A sentence that is either true or false (not both)

  • Truth Value: Whether a statement is true or false

  • Negation: The opposite meaning and truth value of a statement

  • Compound Statement: Two or more statements joined by "and" or "or"

Conditional Statements

  • Structure: "If P, then Q" where P is the hypothesis and Q is the conclusion

  • False only when: Hypothesis is true and conclusion is false

Related Conditionals

  • Converse: Swap hypothesis and conclusion (If Q, then P)

  • Inverse: Negate both parts (If not P, then not Q)

  • Contrapositive: Negate both parts of the converse (If not Q, then not P)

Biconditional Statements

  • Structure: "P if and only if Q"

  • True when: Both parts have the same truth value

  • Combines a conditional with its converse

2. Reasoning Methods

Inductive vs. Deductive Reasoning

Type

Method

Reliability

Example

Inductive

Patterns → General conclusion

Not guaranteed

"All swans I've seen are white, so all swans are white"

Deductive

General rules → Specific conclusion

Guaranteed valid

"All men are mortal. Socrates is a man. Therefore, Socrates is mortal."

Inductive Reasoning Process

  1. Observe patterns in specific examples

  2. Form a conjecture (educated guess)

  3. Test with counterexamples

  4. If no counterexamples found, seek proof

Key: Only one counterexample is needed to disprove a conjecture!

3. Laws of Deductive Reasoning

Law of Detachment

  • If: "If P, then Q" is true AND P is true

  • Then: Q must be true

  • Example: "If it rains, the ground gets wet. It's raining. Therefore, the ground is wet."

Law of Syllogism

  • If: "If P, then Q" AND "If Q, then R" are both true

  • Then: "If P, then R" is true

  • Example: "If I study, I'll pass. If I pass, I'll graduate. Therefore, if I study, I'll graduate."

4. Properties of Equality

Algebraic Properties

  • Addition: If a = b, then a + c = b + c

  • Subtraction: If a = b, then a - c = b - c

  • Multiplication: If a = b, then ac = bc

  • Division: If a = b and c ≠ 0, then a/c = b/c

Fundamental Properties

  • Reflexive: a = a

  • Symmetric: If a = b, then b = a

  • Transitive: If a = b and b = c, then a = c

  • Substitution: If a = b, then b can replace a in any expression

Operational Properties

  • Distributive: a(b + c) = ab + ac

  • Commutative: a + b = b + a or ab = ba

  • Associative: (a + b) + c = a + (b + c)

5. Geometric Postulates

A postulate (axiom) is a statement accepted as true without proof

Postulate

Statement

12.1

Through any two points, there is exactly one line

12.2

Through any three non-collinear points, there is exactly one plane

12.3

A line contains at least two points

12.4

A plane contains at least three non-collinear points

12.5

If two points lie in a plane, then the entire line containing them lies in that plane

12.6

If two lines intersect, they intersect at exactly one point

12.7

If two planes intersect, their intersection is a line

6. Midpoint Theory

Definition vs. Theorem

  • Definition (Equality): If M is midpoint of AB, then AM = MB

  • Theorem (Congruence): If M is midpoint of AB, then AM ≅ MB

Conversion Between Forms

  • Definition → Theorem: Use definition of congruence

  • Within equality mode: Use substitution property of equality

Important: Substitution property of equality can only be used in equality mode!

Practice Questions

Question 1 - Logic

Given the conditional "If a figure is a square, then it has four sides":

  • Write the converse, inverse, and contrapositive

  • Which of these statements is logically equivalent to the original?

Question 2 - Reasoning

Determine whether each conclusion uses inductive or deductive reasoning:

  • "All the dogs I've met are friendly, so all dogs must be friendly"

  • "All mammals have lungs. Whales are mammals. Therefore, whales have lungs"

Question 3 - Laws

Use the Law of Detachment or Syllogism to reach a valid conclusion:

  • "If you exercise regularly, then you'll be healthy. If you're healthy, then you'll live longer. Maya exercises regularly."

Question 4 - Counterexamples

Find a counterexample to disprove: "All prime numbers are odd"

Question 5 - Properties

Name the property that justifies each step:

  • If 3x - 7 = 14, then 3x = 21 (Addition Property)

  • If 3x = 21, then x = 7 (Division Property)

Cross-Document Connections

  • Writing Proofs provides the algebraic foundation for the logical structures in

    Statements, Conditionals, Biconditionals

  • Deductive Reasoning shows how to apply the conditional logic from

    Statements, Conditionals, Biconditionals

  • Conjectures and Counterexamples demonstrates when deductive reasoning is needed instead of inductive reasoning