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Logical Connectives - Intermediate Level: disjunction INTERMEDIATE

Boost your speed and accuracy with this adaptive style 📈 worksheet. Worksheet 15 of 30 presents 20 intermediate-level logical connectives problems. Focus on disjunction while practicing logical operators, AND-OR-NOT logic, conditional statements. Difficulty: moderate complexity with mixed patterns. Perfect for mid-level test takers.

📝 Worksheet 15 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Logical Connectives
Worksheet 15 of 30 (50% complete)

Question 1

Rewrite the following statement in standard 'if-then' form: "The plant will die unless you water it" 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: The plant will die unless you water it
p: The plant will die, q: You water it

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 water it, then the plant will die
unless, conditional, advanced, language

Question 2

Consider the biconditional statement: "p: A number is divisible by 4 if and only if q: The number is even" (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
biconditional, truth-table, equivalence

Question 3

Are the following two logical expressions equivalent? Expression 1: ¬(¬p) Expression 2: p 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)
Expression 2: p

Step 3: Apply Double Negation law
Two negations cancel each other out.
¬(¬p) simply gives back p.
These expressions ARE equivalent.
equivalence, laws, advanced

Question 4

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.
syllogism, all-some-no, categorical, reasoning

Question 5

Consider the biconditional statement: "p: Today is Sunday if and only if q: Tomorrow is Monday" (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
biconditional, truth-table, equivalence

Question 6

Consider the statements: p: Sarah is present q: Sarah is attentive If p is False and q is True, what is the truth value of p ∧ q (p AND q)?
Step 1: Understand the conjunction (AND) operator
The conjunction p ∧ q is True ONLY when BOTH p and q are True.

Step 2: Apply the given values
p = False, q = True

Step 3: Evaluate p ∧ q
Since at least one of p or q is False, p ∧ q = False
conjunction, truth-table, basic

Question 7

Consider the statements: p: Lisa likes coffee q: Lisa likes tea 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
disjunction, truth-table, basic

Question 8

Consider the statement: "Either p: The answer is A OR q: The answer is B, 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
xor, exclusive-or, truth-table

Question 9

Translate this English sentence into symbolic logic: "Either you pay or you leave, but not both." Let p = 'You pay', q = 'You leave' (use appropriate letters).
'Either P or Q, but not both' is exclusive OR: P ⊕ Q.

Answer: p ⊕ q
translation, english-to-logic, symbolic

Question 10

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
biconditional, truth-table, equivalence

Question 11

Consider the statement: p: The number is even If p is False, what is the truth value of ¬p (NOT p)?
Step 1: Understand the negation (NOT) operator
The negation ¬p simply reverses the truth value of p.
If p is True, then ¬p is False.
If p is False, then ¬p is True.

Step 2: Apply the given value
p = False

Step 3: Evaluate ¬p
Since p is False, ¬p = True
In other words: ¬p: The number is odd is True
negation, basic

Question 12

Identify the logical fallacy in this argument: "If you study, you'll pass. You didn't study, so you won't pass." What fallacy is being committed?
Fallacy of denying the antecedent. Form: If P then Q, not P, therefore not Q. You might still pass without studying.
fallacy, reasoning, critical-thinking

Question 13

Consider the relationship between: P: Being divisible by 4 Q: Being an even number Is P a necessary condition, sufficient condition, both, or neither for Q?
Step 1: Understand necessary and sufficient conditions
• P is NECESSARY for Q: Q cannot be true without P (Q → P)
• P is SUFFICIENT for Q: P being true guarantees Q (P → Q)
• P is BOTH: P if and only if Q (P ↔ Q)

Step 2: Analyze the relationship
P: Being divisible by 4
Q: Being an even number

Step 3: Determine the condition type
All numbers divisible by 4 are even (sufficient), but not all even numbers are divisible by 4 (not necessary)

Answer: Sufficient but not necessary
necessary, sufficient, advanced

Question 14

Consider the relationship between: P: Studying Q: Passing the exam Is P a necessary condition, sufficient condition, both, or neither for Q?
Step 1: Understand necessary and sufficient conditions
• P is NECESSARY for Q: Q cannot be true without P (Q → P)
• P is SUFFICIENT for Q: P being true guarantees Q (P → Q)
• P is BOTH: P if and only if Q (P ↔ Q)

Step 2: Analyze the relationship
P: Studying
Q: Passing the exam

Step 3: Determine the condition type
You need to study to pass (necessary), but studying alone doesn't guarantee passing (not sufficient)

Answer: Necessary but not sufficient
necessary, sufficient, advanced

Question 15

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
unless, conditional, advanced, language

Question 16

Consider the statements: p: It is raining q: The ground is wet If p is True and q is False, what is the truth value of p ∧ q (p AND q)?
Step 1: Understand the conjunction (AND) operator
The conjunction p ∧ q is True ONLY when BOTH p and q are True.

Step 2: Apply the given values
p = True, q = False

Step 3: Evaluate p ∧ q
Since at least one of p or q is False, p ∧ q = False
conjunction, truth-table, basic

Question 17

Evaluate this logical argument: Premise: If you study, you will pass. Premise: You studied. Therefore, you will pass. Is this argument valid? (If the premises are true, must the conclusion be true?)
Argument form: Modus Ponens

Modus Ponens: P → Q and P together force Q to be true.

Conclusion: This argument is VALID.
validity, modus-ponens, modus-tollens, fallacy, reasoning

Question 18

Logic puzzle: You meet two people. A says: 'At least one of us is a knave.' B says nothing. What are A, B, and C (or A and B)?
If A knave → statement 'at least one knave' true → but knave can't tell truth → impossible. So A knight → statement true → at least one knave → since A knight, B must be knave.
puzzle, multi-person, advanced, competition

Question 19

Given: p → q, ¬q What can you validly derive?
Rule: Modus Tollens

If P → Q and Q is false, then P must be false.

Answer: ¬p
deduction, proof, inference-rules

Question 20

Evaluate the compound logical expression: (p ∧ q) ∨ r Given: p = True, q = True, r = False
Step 1: Break down the compound expression
Expression: (p ∧ q) ∨ r

Step 2: Evaluate inner expression first
p ∧ q = True ∧ True = True

Step 3: Evaluate outer expression
(True) ∨ False = True
Since OR is True when at least one operand is True
compound, nested, complex
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