Solved Exercises on Logical Reasoning in Grade 7

Master logical reasoning: conditional statements, logical operators, truth tables, and deductive reasoning through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Conditional Statements
Exercise 1
Consider the statement: "If it rains, then the ground gets wet." Identify the hypothesis, conclusion, and write the contrapositive.
Definition:

Conditional Statement: A statement of the form "If p, then q" (written as p → q).

Hypothesis: The "if" part (p), also called the antecedent.

Conclusion: The "then" part (q), also called the consequent.

Conditional statement analysis:
  1. Identify the "if" and "then" parts
  2. Label the hypothesis and conclusion
  3. Form the contrapositive by negating both and reversing
  4. Remember: Original and contrapositive are logically equivalent
Step 1: Identify hypothesis and conclusion

Hypothesis (p): "It rains"

Conclusion (q): "The ground gets wet"

Step 2: Write in logical form

Original: p → q ("If p, then q")

Step 3: Form the contrapositive

Contrapositive: ~q → ~p ("If not q, then not p")

"If the ground does not get wet, then it did not rain."

Step 4: Verify equivalence

The original and contrapositive have the same truth value.

Final answer:

Hypothesis: "It rains", Conclusion: "The ground gets wet", Contrapositive: "If the ground does not get wet, then it did not rain."

Applied rules:

• Conditional structure: If p then q (p → q)

• Contrapositive: ~q → ~p is equivalent to p → q

• Logical equivalence: Original and contrapositive have same truth value

If p then q Hypothesis (p) Conclusion (q) If ~q then ~p Conditional Statement Structure
2 Truth Tables
Exercise 2
Create a truth table for the logical expression: p ∧ (q ∨ r). Then determine when this expression is true.
Definition:

Truth Table: A table showing all possible truth values of logical expressions.

Logical Operators: ∧ (AND), ∨ (OR), ~ (NOT), → (IMPLIES).

Step 1: List all possible combinations

For 3 variables (p, q, r), there are 2³ = 8 combinations

Step 2: Create the truth table
p q r q ∨ r p ∧ (q ∨ r)
T T T T T
T T F T T
T F T T T
T F F F F
F T T T F
F T F T F
F F T T F
F F F F F
Step 3: Analyze the results

p ∧ (q ∨ r) is true only when p is true AND (q ∨ r) is true.

This happens in the first three rows of the table.

Step 4: State the condition

The expression is true when p is true and at least one of q or r is true.

Final answer:

The expression p ∧ (q ∨ r) is true when p is true and at least one of q or r is true (3 out of 8 cases).

Applied rules:

• AND operator (∧): True only when both operands are true

• OR operator (∨): True when at least one operand is true

• Truth table completeness: Must include all 2^n possible combinations

q ∨ r p ∧ (q∨r) Logic Gates for p ∧ (q ∨ r)
3 Deductive Reasoning
Exercise 3
Given: All mammals are warm-blooded. All dogs are mammals. Using deductive reasoning, prove that all dogs are warm-blooded.
Definition:

Deductive Reasoning: Drawing specific conclusions from general principles or premises.

Syllogism: A logical argument with two premises leading to a conclusion.

Step 1: Identify the premises

Premise 1: All mammals are warm-blooded (Mammals → Warm-blooded)

Premise 2: All dogs are mammals (Dogs → Mammals)

Step 2: Apply the syllogistic rule

If A → B and B → C, then A → C

Dogs → Mammals → Warm-blooded

Step 3: Draw the conclusion

Therefore: All dogs are warm-blooded (Dogs → Warm-blooded)

Step 4: Verify the logical structure

The argument follows valid logical form: universal affirmative syllogism

Final answer:

By deductive reasoning, since all mammals are warm-blooded and all dogs are mammals, it follows that all dogs are warm-blooded.

Applied rules:

• Transitivity: If A implies B and B implies C, then A implies C

• Syllogistic reasoning: Drawing conclusions from two premises

• Universal quantifiers: "All" statements apply to entire categories

Warm-blooded Mammals Dogs Dogs Mammals Warm-blooded Deductive Syllogism
Logical Reasoning Concepts, Rules and Methods
p → q ≡ ~q → ~p
Contrapositive Equivalence
AND (∧)
True when both true
Conjunction
OR (∨)
True when at least one true
Disjunction
NOT (~)
Reverses truth value
Negation
Key definitions:

Logical Reasoning: The process of using rational thinking to reach valid conclusions based on given premises.

Proposition: A statement that is either true or false.

Logical Operator: Symbols used to connect propositions (AND, OR, NOT, IMPLIES).

Valid Argument: An argument where the conclusion necessarily follows from the premises.

Logical reasoning methodology:
  1. Identify propositions: Break down statements into basic logical components
  2. Apply operators: Use logical operators to connect propositions
  3. Construct arguments: Build logical chains from premises to conclusion
  4. Verify validity: Check if conclusion follows logically from premises
Tip 1: In "if-then" statements, focus on the logical connection, not causation.
Tip 2: The contrapositive always has the same truth value as the original.
Tip 3: "Or" in logic is inclusive (at least one is true).
Tip 4: Always consider all possible cases when building truth tables.
Key characteristics: Based on formal rules, independent of content, focuses on structure.
Common applications: Computer programming, mathematics, philosophy, law, and scientific reasoning.
Solution: Exercises 4 to 5
4 Logical Puzzles
Exercise 4
In a group of 5 people, A says "B is lying," B says "C is lying," C says "D is lying," D says "E is lying," and E says "A is lying." If exactly one person is telling the truth, who is it?
Definition:

Logical Puzzle: A problem requiring logical deduction to find a unique solution.

Step 1: Analyze the pattern

Each person accuses the next person of lying in a circular pattern: A→B→C→D→E→A

Step 2: Consider the constraint

Exactly one person is telling the truth, the other four are lying.

Step 3: Test each possibility

If A tells the truth: B is lying, so C is telling the truth → contradiction (more than one truth-teller)

If B tells the truth: C is lying, so D is telling the truth → contradiction

If C tells the truth: D is lying, so E is telling the truth → contradiction

If D tells the truth: E is lying, so A is telling the truth → contradiction

If E tells the truth: A is lying, so B is telling the truth → contradiction

Step 4: Reconsider the problem

Wait - if A is lying about B lying, then B is telling the truth. If B is telling the truth about C lying, then C is lying. If C is lying about D lying, then D is telling the truth. If D is telling the truth about E lying, then E is lying. If E is lying about A lying, then A is telling the truth.

This creates A and B both telling the truth → contradiction.

Step 5: Systematic approach

Let's suppose E tells the truth: "A is lying" → A is lying → B is telling the truth → C is lying → D is telling the truth → E is lying. This contradicts our assumption.

Let's suppose D tells the truth: "E is lying" → E is lying → A is telling the truth → B is lying → C is telling the truth → D is lying. This contradicts our assumption.

Let's suppose C tells the truth: "D is lying" → D is lying → E is telling the truth → A is lying → B is telling the truth → C is lying. This contradicts our assumption.

Let's suppose B tells the truth: "C is lying" → C is lying → D is telling the truth → E is lying → A is telling the truth → B is lying. This contradicts our assumption.

Let's suppose A tells the truth: "B is lying" → B is lying → C is telling the truth → D is lying → E is telling the truth → A is lying. This contradicts our assumption.

Step 6: Correct approach

Actually, if exactly one person is telling the truth, and that person is C, then: C tells the truth → D is lying → E is telling the truth. This gives us two truth-tellers, which violates the condition.

Wait, let's reconsider: If D is lying, then E is telling the truth. If E is telling the truth, then A is lying. If A is lying, then B is telling the truth. If B is telling the truth, then C is lying. If C is lying, then D is telling the truth. This creates a contradiction with D lying.

Let's try: B is the truth-teller. B says C is lying → C is lying → D is telling the truth → E is lying → A is telling the truth → B is lying. Contradiction.

Let's try: D is the truth-teller. D says E is lying → E is lying → A is telling the truth → B is lying → C is telling the truth → D is lying. Contradiction.

Let's try: A is the truth-teller. A says B is lying → B is lying → C is telling the truth → D is lying → E is telling the truth → A is lying. Contradiction.

Let's try: C is the truth-teller. C says D is lying → D is lying → E is telling the truth → A is lying → B is telling the truth → C is lying. Contradiction.

Let's try: E is the truth-teller. E says A is lying → A is lying → B is telling the truth → C is lying → D is telling the truth → E is lying. Contradiction.

This seems impossible. Let me re-examine the logic.

If A tells the truth: A says B is lying → B is lying → B's statement "C is lying" is false → C is telling the truth → C's statement "D is lying" is true → D is lying → D's statement "E is lying" is false → E is telling the truth → E's statement "A is lying" is true → A is lying. Contradiction.

Let's try B tells the truth: B says C is lying → C is lying → C's statement "D is lying" is false → D is telling the truth → D's statement "E is lying" is true → E is lying → E's statement "A is lying" is false → A is telling the truth → A's statement "B is lying" is true → B is lying. Contradiction.

Let's try C tells the truth: C says D is lying → D is lying → D's statement "E is lying" is false → E is telling the truth → E's statement "A is lying" is true → A is lying → A's statement "B is lying" is false → B is telling the truth → B's statement "C is lying" is true → C is lying. Contradiction.

Let's try D tells the truth: D says E is lying → E is lying → E's statement "A is lying" is false → A is telling the truth → A's statement "B is lying" is true → B is lying → B's statement "C is lying" is false → C is telling the truth → C's statement "D is lying" is true → D is lying. Contradiction.

Let's try E tells the truth: E says A is lying → A is lying → A's statement "B is lying" is false → B is telling the truth → B's statement "C is lying" is true → C is lying → C's statement "D is lying" is false → D is telling the truth → D's statement "E is lying" is true → E is lying. Contradiction.

All possibilities lead to contradictions. Let me reconsider: perhaps the puzzle is designed so that none of them can be the sole truth-teller under these constraints, which means the puzzle has no solution under the given conditions.

Actually, wait. Let me re-read: "exactly one person is telling the truth". This creates a circular dependency that makes it impossible for exactly one person to be telling the truth. This is a classic logical paradox.

Upon reflection, this puzzle has no solution with exactly one truth-teller due to the circular nature of the accusations.

Final answer:

This puzzle has no solution with exactly one truth-teller due to the circular nature of the accusations creating a logical paradox.

Applied rules:

• Logical consistency: All statements must be consistent with the given constraints

• Systematic testing: Test each possibility to find contradictions

• Paradox recognition: Identify when a problem has no logical solution

A B C D E LOGICAL PARADOX Circular Logic Puzzle
5 Complex Logical Arguments
Exercise 5
Given these premises: (1) If it's sunny, then I'll go to the beach. (2) If I go to the beach, then I'll swim. (3) I didn't swim. Using logical reasoning, what can you conclude?
Definition:

Modus Tollens: A valid logical argument form: If p→q and ~q, then ~p.

Chain of Implications: If p→q and q→r, then p→r.

Step 1: Define the propositions

p: "It's sunny"

q: "I'll go to the beach"

r: "I'll swim"

Step 2: Express the premises

Premise 1: p → q (If sunny, then beach)

Premise 2: q → r (If beach, then swim)

Premise 3: ~r (Not swimming)

Step 3: Apply transitivity

From p → q and q → r, we can deduce p → r

If it's sunny, then I'll swim

Step 4: Apply Modus Tollens

We have p → r and ~r (I didn't swim)

By Modus Tollens: ~p (It's not sunny)

Step 5: Verify the conclusion

If it were sunny, I would have gone to the beach, then swam

Since I didn't swim, I didn't go to the beach

Since I didn't go to the beach, it wasn't sunny

Final answer:

It is not sunny today.

Applied rules:

• Transitivity: If p→q and q→r, then p→r

• Modus Tollens: If p→q and ~q, then ~p

• Logical chain: Build arguments step by step from premises

Sunny Beach Swim ~Swim ~Sunny Logical Argument Chain
Logical Reasoning Theory: Laws, Methods, Definitions, and Formulas
p → q ≡ ~p ∨ q
Conditional as Disjunction
Key definitions:

Logical Reasoning: The systematic process of using rational thinking to reach valid conclusions based on given premises and logical principles.

Proposition: A declarative statement that is either true or false but not both.

Logical Operator: A symbol or word that connects propositions to form compound statements (AND, OR, NOT, IMPLIES).

Argument: A sequence of statements where premises are intended to support a conclusion.

Logical reasoning methodology:
  1. Proposition identification: Recognize and represent statements as logical propositions
  2. Operator application: Use logical operators to connect propositions appropriately
  3. Argument construction: Build logical sequences from premises to conclusions
  4. Validity verification: Check if the conclusion follows necessarily from the premises
  5. Truth table analysis: Use systematic enumeration to verify logical equivalences
Tip 1: In "if-then" statements, focus on the logical relationship, not causation.
Tip 2: The contrapositive is always logically equivalent to the original statement.
Tip 3: "Or" in logic is inclusive (can be both), unlike exclusive "or" in everyday language.
Tip 4: Practice with truth tables to understand how logical operators work.

Key characteristics: Formal system based on rules, independent of content, focuses on structure and validity.
Common applications: Computer science, mathematics, philosophy, legal reasoning, and scientific method.
Essential formulas and rules:

• Conditional: p → q ≡ ~p ∨ q

• Contrapositive: p → q ≡ ~q → ~p

• Converse: p → q → q → p (not equivalent)

• Inverse: p → q → ~p → ~q (not equivalent)

• De Morgan's Laws: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q

Questions & Answers

Question: I'm confused about the difference between "if-then" and "and" statements. When should I use each one?

Answer: These serve different logical purposes:

  • If-then (→): Shows a conditional relationship: "If p, then q" means whenever p is true, q must also be true
  • And (∧): Shows conjunction: "p and q" is true only when both p and q are true simultaneously

Example: "If it rains, then the ground gets wet" (conditional) vs. "It is raining and the ground is wet" (conjunction).

The conditional expresses a cause-effect or dependency relationship, while "and" simply combines two statements.

In a conditional, if p is false, the whole statement is automatically true regardless of q. In a conjunction, if either statement is false, the whole statement is false.

Question: What's the difference between the contrapositive, converse, and inverse of a conditional statement?

Answer: Starting with "If p, then q" (p → q):

  • Contrapositive: "If not q, then not p" (~q → ~p) - Logically equivalent to original
  • Converse: "If q, then p" (q → p) - Not equivalent to original
  • Inverse: "If not p, then not q" (~p → ~q) - Not equivalent to original

The original statement and its contrapositive always have the same truth value.

The converse and inverse have the same truth value as each other but not necessarily the same as the original.

Example: Original: "If it's a dog, then it's a mammal" → Contrapositive: "If it's not a mammal, then it's not a dog" (both true)

Question: How do I know if an argument is valid or invalid?

Answer: An argument is valid if the conclusion necessarily follows from the premises:

  • Valid: If all premises are true, the conclusion must be true
  • Invalid: It's possible for all premises to be true while the conclusion is false

To test validity, assume the premises are true and see if the conclusion must follow.

Example of valid argument: Premises: "All humans are mortal" and "Socrates is human" → Conclusion: "Socrates is mortal"

Example of invalid argument: Premises: "All birds can fly" and "Penguins are birds" → Conclusion: "Penguins can fly" (the first premise is false, but assuming it's true, the conclusion doesn't follow from the premises).

Question: What's the difference between deductive and inductive reasoning?

Answer: These are two different approaches to reasoning:

  • Deductive: Starts with general principles and reaches specific conclusions (top-down). If premises are true, conclusion must be true.
  • Inductive: Starts with specific observations and reaches general conclusions (bottom-up). Conclusion is probably true based on evidence.

Deductive example: "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." (Conclusion necessarily follows)

Inductive example: "Every swan I've seen is white. Therefore, all swans are white." (Conclusion likely but not guaranteed)

Deductive reasoning provides certainty, while inductive reasoning provides probability.

Question: How do I handle logical puzzles when the solution seems impossible?

Answer: Some logical puzzles are designed to have no solution or to create paradoxes:

  • Systematic approach: Test all possibilities to confirm no solution exists
  • Look for contradictions: Identify when assumptions lead to logical inconsistencies
  • Recognize paradoxes: Some puzzles are intentionally unsolvable to teach about logical limitations

Example: "This statement is false" - if true, then false; if false, then true. This is a logical paradox.

When you encounter apparent impossibilities, document your reasoning process. Sometimes the realization that no solution exists is the answer.

In academic settings, most puzzles have solutions, but recognizing when logic breaks down is also valuable.