Geometry Unit 1: Reasoning, Logic & Venn Diagrams

Essential Question for the Unit:

"How can logic be used to justify geometric reasoning?"

Weekly Pacing at a Glance

  • Day 1: Conditional statements and their truth values.
  • Day 2: Converse, inverse, contrapositive, and biconditional statements.
  • Day 3: Symbolic logic (negation, conjunction, disjunction) and compound statements.
  • Day 4: Venn diagrams and their applications in contextual situations.

1. Conditional Statements (If-Then Statements)

A conditional statement is a logical statement with two parts: a hypothesis and a conclusion.

Structure: If [hypothesis], then [conclusion].

Symbolic Form: p → q (Read as "p implies q" or "if p, then q")

Example: "If an angle is a right angle, then its measure is 90°."

  • Hypothesis (p): An angle is a right angle.
  • Conclusion (q): Its measure is 90°.
Type Description Symbolic Form Example
Conditional The original statement. p → q If an angle is a right angle, then it is 90°.
Converse Switch the hypothesis and conclusion. q → p If an angle is 90°, then it is a right angle.
Inverse Negate the original hypothesis and conclusion. ~p → ~q If an angle is not a right angle, then it is not 90°.
Contrapositive Negate AND switch the hypothesis and conclusion. ~q → ~p If an angle is not 90°, then it is not a right angle.

Key Connection: A conditional statement is logically equivalent to its contrapositive. The converse is logically equivalent to the inverse.

2. Symbolic Logic & Compound Statements

Symbol Name Reads As... Meaning
~p Negation "not p" The opposite of statement p.
p ∧ q Conjunction "p and q" True only when BOTH p and q are true.
p ∨ q Disjunction "p or q" True when AT LEAST ONE of p or q is true.
p ↔ q Biconditional "p if and only if q" True only when p and q have the same truth value. (The conditional and converse are both true).

3. Laws of Deductive Reasoning

Deductive reasoning uses facts, definitions, and accepted properties in a logical order to draw a conclusion.

Law of Detachment

If a conditional statement (p → q) is true and its hypothesis (p) is true, then its conclusion (q) must be true.

Example:

1. If it is Friday, then Aly wears jeans. (True conditional)

2. Today is Friday. (Hypothesis is true)

Conclusion: Therefore, Aly is wearing jeans.

Law of Syllogism

If p → q and q → r are true conditional statements, then p → r is also true. (Think of it as a chain reaction).

Example:

1. If you get a driver's license (p), then you must pass a written test (q).

2. If you pass a written test (q), then you have studied the handbook (r).

Conclusion: Therefore, if you get a driver's license (p), then you have studied the handbook (r).

4. Venn Diagrams: Visualizing Set Relationships

Venn diagrams use circles to represent sets and show the relationships between them.

Relationship Description Symbolic
Intersection Elements that are in BOTH set A AND set B. A ∧ B
Union Elements that are in set A OR set B (or both). A ∨ B
Complement Elements that are NOT in set A. ~A
Subset All elements of set A are also in set B. A ⊂ B

Worked Example:

A survey asks students if they like Comedy movies (C) or Horror movies (H). The diagram shows the results.

C
H
8
2
5
  • How many students like Comedy? 8 (only C) + 2 (both) = 10 students.
  • How many students like Horror? 5 (only H) + 2 (both) = 7 students.
  • Comedy AND Horror (Intersection C ∧ H): 2 students.
  • Comedy OR Horror (Union C ∨ H): 8 + 2 + 5 = 15 students.