Logic - Tenth Grade Math
Introduction to Logical Reasoning in Geometry
Logic: The study of reasoning and the principles of valid inference
Statement: A sentence that is either true or false (but not both)
Conditional Statement: A statement in "if-then" form
Purpose in Geometry: To write proofs, make valid conclusions, and understand mathematical relationships
Statement: A sentence that is either true or false (but not both)
Conditional Statement: A statement in "if-then" form
Purpose in Geometry: To write proofs, make valid conclusions, and understand mathematical relationships
1. Identify Hypotheses and Conclusions
Conditional Statement: A statement that can be written in "If-Then" form
Hypothesis: The "if" part of the statement (the condition)
Conclusion: The "then" part of the statement (the result)
Notation: $p \to q$ (read as "if p, then q" or "p implies q")
Hypothesis: The "if" part of the statement (the condition)
Conclusion: The "then" part of the statement (the result)
Notation: $p \to q$ (read as "if p, then q" or "p implies q")
Conditional Statement Structure:
If [hypothesis], then [conclusion]
Symbolic Form: $p \to q$
where:
• $p$ = hypothesis (antecedent)
• $q$ = conclusion (consequent)
• $\to$ = "implies" or "if...then"
If [hypothesis], then [conclusion]
Symbolic Form: $p \to q$
where:
• $p$ = hypothesis (antecedent)
• $q$ = conclusion (consequent)
• $\to$ = "implies" or "if...then"
Example 1: "If two angles are vertical, then they are congruent."
Hypothesis (p): Two angles are vertical
Conclusion (q): They are congruent
Symbolic form: $p \to q$
Hypothesis (p): Two angles are vertical
Conclusion (q): They are congruent
Symbolic form: $p \to q$
Example 2: "If a figure is a square, then it has four equal sides."
Hypothesis: A figure is a square
Conclusion: It has four equal sides
Hypothesis: A figure is a square
Conclusion: It has four equal sides
Example 3: "All right angles measure 90°."
Rewrite in if-then form:
"If an angle is a right angle, then it measures 90°."
Hypothesis: An angle is a right angle
Conclusion: It measures 90°
Rewrite in if-then form:
"If an angle is a right angle, then it measures 90°."
Hypothesis: An angle is a right angle
Conclusion: It measures 90°
Steps to Identify Hypothesis and Conclusion:
Step 1: Rewrite statement in "If...then" form if needed
Step 2: Identify what comes immediately after "if" (hypothesis)
Step 3: Identify what comes immediately after "then" (conclusion)
Step 4: Check that hypothesis is the condition and conclusion is the result
Step 1: Rewrite statement in "If...then" form if needed
Step 2: Identify what comes immediately after "if" (hypothesis)
Step 3: Identify what comes immediately after "then" (conclusion)
Step 4: Check that hypothesis is the condition and conclusion is the result
2. Counterexamples
Counterexample: An example that shows a statement is false
Key Property: Hypothesis is true, but conclusion is false
Purpose: To disprove conjectures or conditional statements
Important: One counterexample is enough to disprove a statement!
Key Property: Hypothesis is true, but conclusion is false
Purpose: To disprove conjectures or conditional statements
Important: One counterexample is enough to disprove a statement!
Counterexample Definition:
For a conditional statement $p \to q$:
A counterexample is a case where:
• The hypothesis ($p$) is TRUE
• BUT the conclusion ($q$) is FALSE
Result: The conditional statement is proven FALSE
For a conditional statement $p \to q$:
A counterexample is a case where:
• The hypothesis ($p$) is TRUE
• BUT the conclusion ($q$) is FALSE
Result: The conditional statement is proven FALSE
Example 1: Statement: "If a figure has four sides, then it is a square."
Counterexample: A rectangle
• A rectangle has four sides (hypothesis TRUE)
• BUT a rectangle is not necessarily a square (conclusion FALSE)
Conclusion: The statement is FALSE
Counterexample: A rectangle
• A rectangle has four sides (hypothesis TRUE)
• BUT a rectangle is not necessarily a square (conclusion FALSE)
Conclusion: The statement is FALSE
Example 2: Statement: "If a number is divisible by 6, then it is divisible by 12."
Counterexample: The number 18
• 18 is divisible by 6 (hypothesis TRUE)
• BUT 18 is not divisible by 12 (conclusion FALSE)
Conclusion: The statement is FALSE
Counterexample: The number 18
• 18 is divisible by 6 (hypothesis TRUE)
• BUT 18 is not divisible by 12 (conclusion FALSE)
Conclusion: The statement is FALSE
Example 3: Statement: "If two angles are supplementary, then they are both acute."
Counterexample: 120° and 60°
• They are supplementary (120° + 60° = 180°) - hypothesis TRUE
• BUT 120° is obtuse, not acute - conclusion FALSE
Conclusion: The statement is FALSE
Counterexample: 120° and 60°
• They are supplementary (120° + 60° = 180°) - hypothesis TRUE
• BUT 120° is obtuse, not acute - conclusion FALSE
Conclusion: The statement is FALSE
Important Notes:
• To prove a statement TRUE: Must work for ALL cases
• To prove a statement FALSE: Need only ONE counterexample
• Not all false statements have obvious counterexamples
• A true statement has NO counterexamples
• To prove a statement TRUE: Must work for ALL cases
• To prove a statement FALSE: Need only ONE counterexample
• Not all false statements have obvious counterexamples
• A true statement has NO counterexamples
3. Conditionals
Conditional Statement: An "if-then" statement with form $p \to q$
Truth Value: Whether a statement is true or false
When TRUE: Whenever hypothesis is true, conclusion must also be true
When FALSE: Only when hypothesis is true AND conclusion is false
Truth Value: Whether a statement is true or false
When TRUE: Whenever hypothesis is true, conclusion must also be true
When FALSE: Only when hypothesis is true AND conclusion is false
Truth Conditions for Conditional $p \to q$:
The conditional is FALSE only when:
• $p$ is TRUE and $q$ is FALSE
The conditional is TRUE when:
• $p$ is TRUE and $q$ is TRUE
• $p$ is FALSE and $q$ is TRUE
• $p$ is FALSE and $q$ is FALSE
Key Insight: "False hypothesis implies anything"
The conditional is FALSE only when:
• $p$ is TRUE and $q$ is FALSE
The conditional is TRUE when:
• $p$ is TRUE and $q$ is TRUE
• $p$ is FALSE and $q$ is TRUE
• $p$ is FALSE and $q$ is FALSE
Key Insight: "False hypothesis implies anything"
Example 1: "If a triangle is equilateral, then it has three equal sides."
This is TRUE because:
• When the hypothesis is true (triangle is equilateral)
• The conclusion is always true (it has three equal sides)
Truth value: TRUE
This is TRUE because:
• When the hypothesis is true (triangle is equilateral)
• The conclusion is always true (it has three equal sides)
Truth value: TRUE
Example 2: "If a quadrilateral is a rectangle, then all its angles are right angles."
Hypothesis: Quadrilateral is a rectangle
Conclusion: All angles are right angles
Truth value: TRUE (by definition of rectangle)
Hypothesis: Quadrilateral is a rectangle
Conclusion: All angles are right angles
Truth value: TRUE (by definition of rectangle)
Example 3: "If 2 + 2 = 5, then circles are square."
Analysis:
• Hypothesis is FALSE (2 + 2 ≠ 5)
• Conclusion is FALSE (circles aren't square)
• But conditional with false hypothesis is considered TRUE!
Truth value: TRUE (vacuously true)
Analysis:
• Hypothesis is FALSE (2 + 2 ≠ 5)
• Conclusion is FALSE (circles aren't square)
• But conditional with false hypothesis is considered TRUE!
Truth value: TRUE (vacuously true)
4. Negations
Negation: The opposite of a statement
Symbol: $\sim p$ or $\neg p$ (read as "not p")
Effect: Changes true to false and false to true
Double Negation: $\sim(\sim p) = p$
Symbol: $\sim p$ or $\neg p$ (read as "not p")
Effect: Changes true to false and false to true
Double Negation: $\sim(\sim p) = p$
Negation Rules:
If $p$ is TRUE, then $\sim p$ is FALSE
If $p$ is FALSE, then $\sim p$ is TRUE
Negating Quantifiers:
• "All" becomes "Not all" or "Some...not"
• "Some" becomes "None" or "No"
• "No" becomes "Some" or "At least one"
If $p$ is TRUE, then $\sim p$ is FALSE
If $p$ is FALSE, then $\sim p$ is TRUE
Negating Quantifiers:
• "All" becomes "Not all" or "Some...not"
• "Some" becomes "None" or "No"
• "No" becomes "Some" or "At least one"
Example 1: Negate simple statements
Statement: "The angle is acute."
Negation: "The angle is not acute."
Statement: $x = 5$
Negation: $x \neq 5$
Statement: "The angle is acute."
Negation: "The angle is not acute."
Statement: $x = 5$
Negation: $x \neq 5$
Example 2: Negate statements with quantifiers
Statement: "All squares are rectangles."
Negation: "Not all squares are rectangles." or "Some squares are not rectangles."
Statement: "Some triangles are equilateral."
Negation: "No triangles are equilateral."
Statement: "All squares are rectangles."
Negation: "Not all squares are rectangles." or "Some squares are not rectangles."
Statement: "Some triangles are equilateral."
Negation: "No triangles are equilateral."
Example 3: Negate compound statements
Statement: "The figure is a circle and it is red."
Negation: "The figure is not a circle or it is not red."
Note: "and" becomes "or" when negating
Statement: "The figure is a circle and it is red."
Negation: "The figure is not a circle or it is not red."
Note: "and" becomes "or" when negating
De Morgan's Laws for Negations:
• $\sim(p \text{ and } q) = \sim p \text{ or } \sim q$
• $\sim(p \text{ or } q) = \sim p \text{ and } \sim q$
• $\sim(p \text{ and } q) = \sim p \text{ or } \sim q$
• $\sim(p \text{ or } q) = \sim p \text{ and } \sim q$
5. Converses, Inverses, and Contrapositives
Related Conditionals: Four related conditional statements
Original: $p \to q$
Converse: $q \to p$ (switch hypothesis and conclusion)
Inverse: $\sim p \to \sim q$ (negate both parts)
Contrapositive: $\sim q \to \sim p$ (switch AND negate)
Original: $p \to q$
Converse: $q \to p$ (switch hypothesis and conclusion)
Inverse: $\sim p \to \sim q$ (negate both parts)
Contrapositive: $\sim q \to \sim p$ (switch AND negate)
Four Forms of Conditional Statements:
Given original: $p \to q$ "If $p$, then $q$"
Converse: $q \to p$
"If $q$, then $p$"
Method: Switch hypothesis and conclusion
Inverse: $\sim p \to \sim q$
"If not $p$, then not $q$"
Method: Negate both hypothesis and conclusion
Contrapositive: $\sim q \to \sim p$
"If not $q$, then not $p$"
Method: Switch and negate both parts
Given original: $p \to q$ "If $p$, then $q$"
Converse: $q \to p$
"If $q$, then $p$"
Method: Switch hypothesis and conclusion
Inverse: $\sim p \to \sim q$
"If not $p$, then not $q$"
Method: Negate both hypothesis and conclusion
Contrapositive: $\sim q \to \sim p$
"If not $q$, then not $p$"
Method: Switch and negate both parts
Example 1: Form all related conditionals
Conditional: "If a figure is a square, then it has four equal sides."
Converse: "If a figure has four equal sides, then it is a square."
(FALSE - could be a rhombus)
Inverse: "If a figure is not a square, then it does not have four equal sides."
(FALSE - could be a rhombus)
Contrapositive: "If a figure does not have four equal sides, then it is not a square."
(TRUE - by logic)
Conditional: "If a figure is a square, then it has four equal sides."
Converse: "If a figure has four equal sides, then it is a square."
(FALSE - could be a rhombus)
Inverse: "If a figure is not a square, then it does not have four equal sides."
(FALSE - could be a rhombus)
Contrapositive: "If a figure does not have four equal sides, then it is not a square."
(TRUE - by logic)
Example 2: Another example
Conditional: "If two angles are vertical, then they are congruent."
Converse: "If two angles are congruent, then they are vertical."
(FALSE)
Inverse: "If two angles are not vertical, then they are not congruent."
(FALSE)
Contrapositive: "If two angles are not congruent, then they are not vertical."
(TRUE)
Conditional: "If two angles are vertical, then they are congruent."
Converse: "If two angles are congruent, then they are vertical."
(FALSE)
Inverse: "If two angles are not vertical, then they are not congruent."
(FALSE)
Contrapositive: "If two angles are not congruent, then they are not vertical."
(TRUE)
Logical Equivalences:
• Conditional and Contrapositive are ALWAYS logically equivalent
$p \to q \equiv \sim q \to \sim p$
• Converse and Inverse are ALWAYS logically equivalent
$q \to p \equiv \sim p \to \sim q$
Important: If original is true, contrapositive is true
Note: Converse may or may not be true
• Conditional and Contrapositive are ALWAYS logically equivalent
$p \to q \equiv \sim q \to \sim p$
• Converse and Inverse are ALWAYS logically equivalent
$q \to p \equiv \sim p \to \sim q$
Important: If original is true, contrapositive is true
Note: Converse may or may not be true
6. Biconditionals
Biconditional Statement: A statement that works both ways
Form: "p if and only if q"
Symbol: $p \leftrightarrow q$ (double arrow)
Abbreviation: "iff" means "if and only if"
Meaning: Both $p \to q$ AND $q \to p$ are true
Form: "p if and only if q"
Symbol: $p \leftrightarrow q$ (double arrow)
Abbreviation: "iff" means "if and only if"
Meaning: Both $p \to q$ AND $q \to p$ are true
Biconditional Definition:
$$p \leftrightarrow q \text{ means } (p \to q) \text{ AND } (q \to p)$$
In words: "$p$ if and only if $q$"
Truth Condition:
$p \leftrightarrow q$ is TRUE when:
• Both $p$ and $q$ are TRUE, OR
• Both $p$ and $q$ are FALSE
$p \leftrightarrow q$ is FALSE when:
• $p$ and $q$ have different truth values
$$p \leftrightarrow q \text{ means } (p \to q) \text{ AND } (q \to p)$$
In words: "$p$ if and only if $q$"
Truth Condition:
$p \leftrightarrow q$ is TRUE when:
• Both $p$ and $q$ are TRUE, OR
• Both $p$ and $q$ are FALSE
$p \leftrightarrow q$ is FALSE when:
• $p$ and $q$ have different truth values
Example 1: Identify biconditionals
Statement: "Two angles are congruent if and only if they have the same measure."
This means BOTH:
• If two angles are congruent, then they have the same measure (TRUE)
• If two angles have the same measure, then they are congruent (TRUE)
Since both directions are true, the biconditional is TRUE
Statement: "Two angles are congruent if and only if they have the same measure."
This means BOTH:
• If two angles are congruent, then they have the same measure (TRUE)
• If two angles have the same measure, then they are congruent (TRUE)
Since both directions are true, the biconditional is TRUE
Example 2: Write as biconditional
Given: "A triangle is equilateral if and only if all three sides are equal."
Forward direction: If a triangle is equilateral, then all three sides are equal
Backward direction: If all three sides are equal, then the triangle is equilateral
Both are true, so biconditional is TRUE
Given: "A triangle is equilateral if and only if all three sides are equal."
Forward direction: If a triangle is equilateral, then all three sides are equal
Backward direction: If all three sides are equal, then the triangle is equilateral
Both are true, so biconditional is TRUE
Example 3: Test if statement can be biconditional
Statement: "A figure is a square if and only if it has four sides."
Forward: If a figure is a square, then it has four sides (TRUE)
Backward: If a figure has four sides, then it is a square (FALSE - could be rectangle)
Biconditional is FALSE because backward direction is false
Statement: "A figure is a square if and only if it has four sides."
Forward: If a figure is a square, then it has four sides (TRUE)
Backward: If a figure has four sides, then it is a square (FALSE - could be rectangle)
Biconditional is FALSE because backward direction is false
Creating Biconditionals:
A biconditional can only be formed when:
• The conditional $p \to q$ is TRUE
• AND the converse $q \to p$ is TRUE
Test: Both directions must be true for biconditional to be true
A biconditional can only be formed when:
• The conditional $p \to q$ is TRUE
• AND the converse $q \to p$ is TRUE
Test: Both directions must be true for biconditional to be true
7. Truth Tables
Truth Table: A table showing all possible truth values of a statement
T: True
F: False
Purpose: To analyze logical statements systematically
Rows: All possible combinations of truth values
T: True
F: False
Purpose: To analyze logical statements systematically
Rows: All possible combinations of truth values
Basic Truth Tables
Truth Table for Negation ($\sim p$):
| $p$ | $\sim p$ |
|---|---|
| T | F |
| F | T |
Truth Table for Conditional ($p \to q$):
Remember: Only FALSE when hypothesis is TRUE and conclusion is FALSE
| $p$ | $q$ | $p \to q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Remember: Only FALSE when hypothesis is TRUE and conclusion is FALSE
Truth Table for Biconditional ($p \leftrightarrow q$):
Remember: TRUE only when both have same truth value
| $p$ | $q$ | $p \leftrightarrow q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Remember: TRUE only when both have same truth value
Complete Truth Table: Related Conditionals
Truth Table Showing All Related Statements:
Notice: Conditional and Contrapositive have identical truth values!
Notice: Converse and Inverse have identical truth values!
| $p$ | $q$ | $\sim p$ | $\sim q$ | $p \to q$ (Conditional) | $q \to p$ (Converse) | $\sim p \to \sim q$ (Inverse) | $\sim q \to \sim p$ (Contrapositive) |
|---|---|---|---|---|---|---|---|
| T | T | F | F | T | T | T | T |
| T | F | F | T | F | T | T | F |
| F | T | T | F | T | F | F | T |
| F | F | T | T | T | T | T | T |
Notice: Conditional and Contrapositive have identical truth values!
Notice: Converse and Inverse have identical truth values!
8. Truth Values
Truth Value: Whether a statement is true (T) or false (F)
Determining Truth: Using definitions, postulates, theorems, or given information
Application: Essential for proofs and logical reasoning in geometry
Determining Truth: Using definitions, postulates, theorems, or given information
Application: Essential for proofs and logical reasoning in geometry
Example 1: Determine truth values
Statement: "If a triangle has three equal sides, then it is equilateral."
Truth value: TRUE (by definition)
Statement: "If a figure is a rectangle, then it is a square."
Truth value: FALSE (counterexample: non-square rectangle)
Statement: "If a triangle has three equal sides, then it is equilateral."
Truth value: TRUE (by definition)
Statement: "If a figure is a rectangle, then it is a square."
Truth value: FALSE (counterexample: non-square rectangle)
Example 2: Using truth table to determine truth value
Given: $p$ is TRUE, $q$ is FALSE
Find truth value of: $p \to q$
From truth table:
When $p$ = T and $q$ = F, then $p \to q$ = F
Answer: FALSE
Given: $p$ is TRUE, $q$ is FALSE
Find truth value of: $p \to q$
From truth table:
When $p$ = T and $q$ = F, then $p \to q$ = F
Answer: FALSE
Example 3: Compound statement truth values
Given: $p$ is TRUE, $q$ is FALSE, $r$ is TRUE
Find: Truth value of $(p \to q) \to r$
Step 1: Find $p \to q$
$p \to q$ = T → F = F
Step 2: Find $(p \to q) \to r$
F → T = T
Answer: TRUE
Given: $p$ is TRUE, $q$ is FALSE, $r$ is TRUE
Find: Truth value of $(p \to q) \to r$
Step 1: Find $p \to q$
$p \to q$ = T → F = F
Step 2: Find $(p \to q) \to r$
F → T = T
Answer: TRUE
Complete Summary of Related Conditionals
| Type | Symbolic Form | How to Form | Example |
|---|---|---|---|
| Conditional | $p \to q$ | Original statement | If square, then four equal sides |
| Converse | $q \to p$ | Switch p and q | If four equal sides, then square |
| Inverse | $\sim p \to \sim q$ | Negate both | If not square, then not four equal sides |
| Contrapositive | $\sim q \to \sim p$ | Switch and negate | If not four equal sides, then not square |
Logical Equivalences
| Statements | Are They Equivalent? | Why? |
|---|---|---|
| Conditional and Contrapositive | YES | Always have same truth value |
| Converse and Inverse | YES | Always have same truth value |
| Conditional and Converse | NO | May have different truth values |
| Conditional and Inverse | NO | May have different truth values |
Quick Reference: Symbols and Notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| $p, q, r$ | Statement variables | Represent statements | $p$ = "It is raining" |
| $\sim$ or $\neg$ | Negation | Not | $\sim p$ = "It is not raining" |
| $\to$ | Conditional | If...then / Implies | $p \to q$ = "If p, then q" |
| $\leftrightarrow$ | Biconditional | If and only if | $p \leftrightarrow q$ = "p iff q" |
| $\equiv$ | Logical equivalence | Logically equivalent | $p \to q \equiv \sim q \to \sim p$ |
| T | True | Statement is true | $2 + 2 = 4$ is T |
| F | False | Statement is false | $2 + 2 = 5$ is F |
Decision Tree: Writing Related Conditionals
| To Form... | Action | Result |
|---|---|---|
| Converse | Switch hypothesis and conclusion | $q \to p$ |
| Inverse | Negate hypothesis AND conclusion | $\sim p \to \sim q$ |
| Contrapositive | Switch AND negate both | $\sim q \to \sim p$ |
| Biconditional | Both directions must be true | $p \leftrightarrow q$ |
Success Tips for Logic in Geometry:
✓ Hypothesis comes after "IF"; Conclusion comes after "THEN"
✓ One counterexample is enough to disprove a statement
✓ Conditional is false ONLY when hypothesis is true and conclusion is false
✓ Conditional and contrapositive are ALWAYS logically equivalent
✓ Converse and inverse are ALWAYS logically equivalent
✓ To form contrapositive: switch AND negate both parts
✓ Biconditional is true when both statements have same truth value
✓ Use truth tables to verify logical equivalences
✓ "If and only if" means both directions must be true
✓ Practice rewriting statements in symbolic form!
✓ Hypothesis comes after "IF"; Conclusion comes after "THEN"
✓ One counterexample is enough to disprove a statement
✓ Conditional is false ONLY when hypothesis is true and conclusion is false
✓ Conditional and contrapositive are ALWAYS logically equivalent
✓ Converse and inverse are ALWAYS logically equivalent
✓ To form contrapositive: switch AND negate both parts
✓ Biconditional is true when both statements have same truth value
✓ Use truth tables to verify logical equivalences
✓ "If and only if" means both directions must be true
✓ Practice rewriting statements in symbolic form!