Question 1
In set theory, what logical connective matches this concept?
The intersection of sets A and B
Intersection means element is in BOTH sets, which is logical AND.
Answer: x ∈ A ∧ x ∈ B
Answer: x ∈ A ∧ x ∈ B
Logical Connectives - Beginner Level: conditional statements BEGINNER
Exam-focused quick revision round worksheet: 20 beginner-level logical connectives questions. Worksheet 3 of 30 targets conditional statements. Build proficiency in conditional statements, logical connectives, propositional connectives with detailed solutions. Ideal for entry-level competitive exam preparation.
What you'll learn in this worksheet:
- Achieve proficiency in conditional statements by completing this worksheet
- Develop strong logical connectives skills with practical problems
- Improve your logical connectives solving speed through quick revision round
- Gain confidence in identifying propositional connectives patterns
- Master real-world applications of conditional statements
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Worksheet 3 of 30 (10% complete)
Question 2
Consider the statement: 'Neither John nor Mary came'
If p is F and q is T, what is the truth value of 'Neither John nor Mary came'?
Step 1: Understand 'neither...nor' statements
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q
Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T
Step 3: Apply given values
p = F, q = T
¬p = T, ¬q = F
¬p ∧ ¬q = F
Answer: False
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q
Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T
Step 3: Apply given values
p = F, q = T
¬p = T, ¬q = F
¬p ∧ ¬q = F
Answer: False
Question 3
Given: p → q, p
What can you validly derive?
Rule: Modus Ponens (→-elimination)
If P → Q and P are true, Q must be true.
Answer: q
If P → Q and P are true, Q must be true.
Answer: q
Question 4
Are the following two logical expressions equivalent?
Expression 1: p → q
Expression 2: ¬p ∨ q
Answer Yes or No and explain why.
Step 1: Understand what logical equivalence means
Two expressions are logically equivalent if they have the same truth value for ALL possible combinations of variables.
Step 2: Analyze the expressions
Expression 1: p → q
Expression 2: ¬p ∨ q
Step 3: Apply Implication equivalence
The implication p → q is equivalent to ¬p ∨ q
This is a fundamental law in logic.
These expressions ARE equivalent.
Two expressions are logically equivalent if they have the same truth value for ALL possible combinations of variables.
Step 2: Analyze the expressions
Expression 1: p → q
Expression 2: ¬p ∨ q
Step 3: Apply Implication equivalence
The implication p → q is equivalent to ¬p ∨ q
This is a fundamental law in logic.
These expressions ARE equivalent.
Question 5
Consider the conditional statement:
"If p: It rains, then q: The match is cancelled"
(p → q)
If p is True and q is False, what is the truth value of p → q?
Step 1: Understand the implication (→) operator
The implication p → q is False ONLY when p is True but q is False.
In all other cases, it is True.
Truth table for p → q:
p=T, q=T → Result=T
p=T, q=F → Result=F (the only False case)
p=F, q=T → Result=T
p=F, q=F → Result=T
Step 2: Apply the given values
p = True, q = False
Step 3: Evaluate p → q
Since p is True but q is False, the implication fails: p → q = False
The implication p → q is False ONLY when p is True but q is False.
In all other cases, it is True.
Truth table for p → q:
p=T, q=T → Result=T
p=T, q=F → Result=F (the only False case)
p=F, q=T → Result=T
p=F, q=F → Result=T
Step 2: Apply the given values
p = True, q = False
Step 3: Evaluate p → q
Since p is True but q is False, the implication fails: p → q = False
Question 6
Consider the statement:
"Either p: He will come today OR q: He will come tomorrow, but NOT both"
If p is False and q is True, is this statement true?
Step 1: Understand Exclusive OR (XOR)
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.
Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)
Step 2: Apply the given values
p = False, q = True
Step 3: Evaluate the exclusive OR
Since exactly one of p or q is True, the exclusive OR is True
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.
Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)
Step 2: Apply the given values
p = False, q = True
Step 3: Evaluate the exclusive OR
Since exactly one of p or q is True, the exclusive OR is True
Question 7
Given: p
What can you validly derive?
Rule: Disjunction Introduction (∨-intro)
If P is true, then P ∨ Q is true for any Q (addition rule).
Answer: p ∨ q
If P is true, then P ∨ Q is true for any Q (addition rule).
Answer: p ∨ q
Question 8
Consider the statement: 'The solution is neither hot nor cold'
If p is F and q is T, what is the truth value of 'The solution is neither hot nor cold'?
Step 1: Understand 'neither...nor' statements
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q
Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T
Step 3: Apply given values
p = F, q = T
¬p = T, ¬q = F
¬p ∧ ¬q = F
Answer: False
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q
Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T
Step 3: Apply given values
p = F, q = T
¬p = T, ¬q = F
¬p ∧ ¬q = F
Answer: False
Question 9
Consider the statement:
"Either p: The answer is A OR q: The answer is B, but NOT both"
If p is True and q is True, is this statement true?
Step 1: Understand Exclusive OR (XOR)
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.
Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)
Step 2: Apply the given values
p = True, q = True
Step 3: Evaluate the exclusive OR
Since both have the same truth value, the exclusive OR is False
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.
Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)
Step 2: Apply the given values
p = True, q = True
Step 3: Evaluate the exclusive OR
Since both have the same truth value, the exclusive OR is False
Question 10
Consider the statements:
p: You can pay by cash
q: You can pay by card
If p is False and q is True, what is the truth value of p ∨ q (p OR q)?
Step 1: Understand the disjunction (OR) operator
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = False, q = True
Step 3: Evaluate p ∨ q
Since at least one of p or q is True, p ∨ q = True
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = False, q = True
Step 3: Evaluate p ∨ q
Since at least one of p or q is True, p ∨ q = True
Question 11
Rewrite the following statement in standard 'if-then' form:
"She won't come unless you invite her"
What is the equivalent conditional statement?
Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'
Step 2: Identify components
Original: She won't come unless you invite her
p: She won't come, q: You invite her
Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p
Step 4: Write in English
Equivalent statement: If you do not invite her, then she won't come
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'
Step 2: Identify components
Original: She won't come unless you invite her
p: She won't come, q: You invite her
Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p
Step 4: Write in English
Equivalent statement: If you do not invite her, then she won't come
Question 12
Rewrite the following statement in standard 'if-then' form:
"You will fail unless you study"
What is the equivalent conditional statement?
Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'
Step 2: Identify components
Original: You will fail unless you study
p: You will fail, q: You study
Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p
Step 4: Write in English
Equivalent statement: If you do not study, then you will fail
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'
Step 2: Identify components
Original: You will fail unless you study
p: You will fail, q: You study
Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p
Step 4: Write in English
Equivalent statement: If you do not study, then you will fail
Question 13
Evaluate this logical argument:
Premise: If it rains, the ground gets wet.
Premise: It is raining.
Therefore, the ground gets wet.
Is this argument valid? (If the premises are true, must the conclusion be true?)
Argument form: Modus Ponens
This is Modus Ponens: If P → Q and P are true, then Q must be true.
Conclusion: This argument is VALID.
This is Modus Ponens: If P → Q and P are true, then Q must be true.
Conclusion: This argument is VALID.
Question 14
Find a counterexample to show this statement is FALSE:
"p → q is logically equivalent to q → p"
Provide truth values for p, q, r that make the two sides different.
Implication is not symmetric. Converse is not equivalent to original.
p=True, q=False: p→q=False, q→p=True → Different!
p=True, q=False: p→q=False, q→p=True → Different!
Question 15
Consider the biconditional statement:
"p: Triangle ABC is equilateral if and only if q: All angles of triangle ABC are 60°"
(p ↔ q)
If p is True and q is True, what is the truth value of p ↔ q?
Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.
Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)
Step 2: Apply the given values
p = True, q = True
Step 3: Evaluate p ↔ q
Since p and q have the same truth value (True), p ↔ q = True
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.
Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)
Step 2: Apply the given values
p = True, q = True
Step 3: Evaluate p ↔ q
Since p and q have the same truth value (True), p ↔ q = True
Question 16
Consider this syllogism:
Premise 1: Some politicians are honest.
Premise 2: No honest people lie.
Therefore, some politicians do not lie.
Is this syllogism logically valid?
Valid: Some A are B, no B are C → Some A are not C.
Question 17
Given the conditional statement:
"If you study hard, then you will pass"
(p → q)
What is the Contrapositive of this statement?
Step 1: Understand the original statement
Original: p → q means "If you study hard, then you will pass"
Step 2: Understand Contrapositive
Contrapositive switches AND negates both parts: ¬q → ¬p
If the original is p → q, the contrapositive is ¬q → ¬p
Important: A conditional and its contrapositive are LOGICALLY EQUIVALENT
Step 3: Apply to our statement
Contrapositive: If you will pass is false, then you study hard is false
Original: p → q means "If you study hard, then you will pass"
Step 2: Understand Contrapositive
Contrapositive switches AND negates both parts: ¬q → ¬p
If the original is p → q, the contrapositive is ¬q → ¬p
Important: A conditional and its contrapositive are LOGICALLY EQUIVALENT
Step 3: Apply to our statement
Contrapositive: If you will pass is false, then you study hard is false
Question 18
Consider the statements:
p: The flight departs at 9 AM
q: The flight departs at 2 PM
If p is True and q is False, what is the truth value of p ∨ q (p OR q)?
Step 1: Understand the disjunction (OR) operator
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = True, q = False
Step 3: Evaluate p ∨ q
Since at least one of p or q is True, p ∨ q = True
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = True, q = False
Step 3: Evaluate p ∨ q
Since at least one of p or q is True, p ∨ q = True
Question 19
Identify the logical fallacy in this argument:
"Either you support our policy, or you hate the country."
What fallacy is being committed?
False dilemma (either/or fallacy). Presents only two options when more exist (e.g., you could disagree but still love your country).
Question 20
Consider the statements:
p: Tom is a doctor
q: Tom is an engineer
If p is False and q is False, what is the truth value of p ∨ q (p OR q)?
Step 1: Understand the disjunction (OR) operator
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = False, q = False
Step 3: Evaluate p ∨ q
Since both p and q are False, p ∨ q = False
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.
Step 2: Apply the given values
p = False, q = False
Step 3: Evaluate p ∨ q
Since both p and q are False, p ∨ q = False
🎯 Stay focused! Worksheet 3 targets conditional statements.