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Logical Connectives - Beginner-Intermediate Level: logical combinations BEGINNER-INTERMEDIATE

Intensive quick response training 🎯 drill: 20 beginner-intermediate-level logical connectives questions. Worksheet 10 of 30 hones your logical combinations abilities. Practice conditional statements, logical connectives, propositional connectives under timed conditions. Best for developing students seeking building on fundamentals with moderate challenges.

📝 Worksheet 10 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner-intermediate level

What you'll learn in this worksheet:
Your progress through Logical Connectives
Worksheet 10 of 30 (33% complete)

Question 1

Consider the statements: p: The car has fuel q: The car engine is working 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 2

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 3

Consider the statement: p: The number is even If p is True, 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 = True

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

Question 4

Complete the truth table for the expression: p → (q ∨ r) What is the truth value when p=F, q=T, r=T?
Step 1: Break down the expression
Expression: p → (q ∨ r)
Given: p=F, q=T, r=T

Step 2: Evaluate inner expressions first
q ∨ r = T ∨ T = T

Step 3: Evaluate outer expression
p → (T) = F → T = True
Remember: Implication is False only when p is True and consequent is False
truth-table, evaluation, complex

Question 5

Complete the truth table for the expression: (p ∧ q) → r What is the truth value when p=F, q=T, r=T?
Step 1: Break down the expression
Expression: (p ∧ q) → r
Given: p=F, q=T, r=T

Step 2: Evaluate inner expressions first
p ∧ q = F ∧ T = F

Step 3: Evaluate outer expression
(F) → T = True
Remember: Implication is False only when antecedent is True and consequent is False
truth-table, evaluation, complex

Question 6

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!
counterexample, equivalence, disproof

Question 7

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 8

Consider the statement: "Either p: He will come today OR q: He will come tomorrow, but NOT both" If p is True and q is False, 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 = False

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

Identify the logical fallacy in this argument: "My opponent argues for more police funding, but he was arrested for tax evasion, so his argument is wrong." What fallacy is being committed?
Attacking the person instead of addressing the argument.
fallacy, reasoning, critical-thinking

Question 10

Given the conditional statement: "If you study hard, then you will pass" (p → q) What is the Converse of this statement?
Step 1: Understand the original statement
Original: p → q means "If you study hard, then you will pass"

Step 2: Understand Converse
Converse switches the hypothesis and conclusion: q → p
If the original is p → q, the converse is q → p

Step 3: Apply to our statement
Converse: If you will pass, then you study hard
transformations, conditional, advanced

Question 11

Consider this syllogism: Premise 1: All birds can fly. Premise 2: Penguins are birds. Therefore, penguins can fly. Is this syllogism logically valid?
This is valid in form, but the premise 'All birds can fly' is false. Validity is about logical structure, not factual truth. Form: All A are B, C is A → C is B.
syllogism, all-some-no, categorical, reasoning

Question 12

Identify the logical fallacy in this argument: "If we allow students to redo tests, next they'll want to rewrite all exams, then abolish grades entirely!" What fallacy is being committed?
Assumes without evidence that one small step leads to extreme consequences.
fallacy, reasoning, critical-thinking

Question 13

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 14

Evaluate this logical argument: Premise: If you study, you will pass. Premise: You did NOT pass. Therefore, you did NOT study. Is this argument valid? (If the premises are true, must the conclusion be true?)
Argument form: Modus Tollens

Modus Tollens: If P → Q and ¬Q, then ¬P follows necessarily.

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

Question 15

Consider the statement: p: She is happy 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: She is not happy is True
negation, basic

Question 16

Given the conditional statement: "If a number is divisible by 4, then it is even" (p → q) What is the Contrapositive of this statement?
Step 1: Understand the original statement
Original: p → q means "If a number is divisible by 4, then it is even"

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 it is even is false, then a number is divisible by 4 is false
transformations, conditional, advanced

Question 17

Consider the statement: p: The door is locked If p is True, 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 = True

Step 3: Evaluate ¬p
Since p is True, ¬p = False
In other words: ¬p: The door is not locked is False
negation, basic

Question 18

Identify the logical fallacy in this argument: "If we allow students to redo tests, next they'll want to rewrite all exams, then abolish grades entirely!" What fallacy is being committed?
Assumes without evidence that one small step leads to extreme consequences.
fallacy, reasoning, critical-thinking

Question 19

Logic puzzle: Three people, A, B, and C, are each either a knight (always tells truth) or knave (always lies). A says: 'B is a knave.' B says: 'A and C are the same type.' C says: 'A is a knight.' What are A, B, and C (or A and B)?
Case analysis: If A knight → B knave (A's truth) → A and C different (B's lie) → C knight? But C says 'A is knight' which would be true, consistent. Wait, need full check.

Actually solve: Assume A knight → 'B knave' true → B knave → B's statement 'A and C same' is false → A and C different → C knave → C says 'A knight' which is false (since A knight?) Contradiction.

Therefore A knave → 'B knave' false → B knight → B's statement true → A and C same → C knave → C says 'A knight' false (since A knave) ✓. Solution: A knave, B knight, C knave.
puzzle, multi-person, advanced, competition

Question 20

In set theory, what logical connective matches this concept? Elements NOT in set A
Complement means element is NOT in the set, which is logical NOT.

Answer: ¬(x ∈ A)
sets, venn, visual, advanced
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