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Total questions: 150
Worksheet time: 1hrs 15mins
Which statement best defines a proposition in mathematical logic?
A declarative sentence with a truth value
Any question with unknown information
A command that directs an action
A variable describing numerical data
Which symbols typically denote truth values for propositions?
+ for true, − for false
T for true, F for false
Yes for truth, No for falsity
1 for false, 0 for true
Which of the following is NOT a proposition?
Seven is a prime number
Close the window now
32+42=72
√2 is irrational
What is the primary purpose of logical rules in mathematics?
Distinguish valid and invalid arguments
Compute large numerical datasets
Graph complex functions precisely
Memorize steps without reasoning
Which field benefits from logic through circuit design and program verification?
Computer science
Organic chemistry
Environmental science
Classical mechanics
Identify the correct truth values for the following: 1) Bangalore is the capital of India, 2) There is a prime number, 3) 2 is irrational, 4) 2+3=5 , 5) 32+42=72
T, F, T, F, F
F, F, T, T, T
T, T, F, T, F
F, T, T, F, F
Which notation is commonly used to denote propositions?
Numerals 1, 2, 3, 4
Greek letters α, β, γ, δ
Uppercase letters A, B, C, D
Lowercase letters p, q, r, s
Why can a proposition not be both true and false at the same time?
Need for computational efficiency
Law of non-contradiction in logic
Preference for simpler notation
Historical convention from Euclid
Which statement is a valid proposition about numbers?
There exists a prime number
Please define a prime number
Is seven actually prime?
Prove primes are infinite
Which application of logic is mentioned as being automated by software systems?
Rendering 3D graphics smoothly
Generating random passwords
Sorting large datasets quickly
Constructing proofs automatically
Which sentence is a proposition in mathematical logic?
Good Morning greeting with exclamation
What time is it question form
Read this carefully imperative
x + 1 = 2 equality statement
Which connective corresponds to conjunction in symbolic logic?
↔ biconditional operator
∧ conjunction operator
→ conditional operator
∨ disjunction operator
Choose the correct symbol for biconditional (iff).
→ right arrow symbol
∧ wedge symbol
↔ double arrow symbol
∨ vee symbol
Negation of p is denoted by which notation?
p with a star postfix p*
p with a plus sign p+
p with a caret superscript p
p with a tilde prefix ~p
If p is true, what is the truth value of ¬p?
True remains unchanged
False becomes true
True becomes false
Undefined without context
Which statement correctly defines compound propositions?
Constructed using logical connectives
Created by quantifiers exclusively
Formed by single variables only
Derived from arithmetic operations
Identify the logical connective used in the phrase “If … then”.
Conjunction connective and
Disjunction connective or
Conditional connective →
Negation connective not
Which of the following is NOT a proposition?
Equality between two numbers
Equation x + 1 = 2
Question asking the time
Statement about Linux
Given p: “Michael’s PC runs Linux”, what is a correct negation?
Michael’s PC definitely runs Linux
Michael’s PC might run Linux
Michael’s PC does not run Linux
Michael’s PC sometimes runs Linux
Match each connective to its name.
∨ to Conditional
∧ to Conjunction
↔ to Negation
→ to Disjunction
Which statement best defines a conjunction p ∧ q?
True only if both are true
True if at least one is true
True when exactly one is true
True unless both are false
Given p: "Rebecca’s PC has more than 16 GB free hard disk space" and q: "The processor runs faster than 1 GHz," what does p ∧ q state?
Only the free space condition holds
Only the processor condition holds
At least one condition must hold
Both free space and processor condition hold
Which row makes p ∧ q true in the truth table?
p=T, q=T
p=F, q=F
p=F, q=T
p=T, q=F
Which statement best defines a disjunction p ∨ q?
False when exactly one is false
True when exactly one is true
True only if both are true
False only if both are false
Which row makes p ∨ q false in the truth table?
p=T, q=F
p=F, q=F
p=T, q=T
p=F, q=T
If q: "Vandana’s smartphone has at least 32GB of memory," which is a correct negation ~q?
Vandana’s smartphone has more than 32GB
Vandana’s smartphone has at least 32GB
Vandana’s smartphone has less than 32GB
Vandana’s smartphone has exactly 32GB
Choose the correct symbolic form of: "p or q".
p ∧ q
~(p ∧ q)
p ∨ q
p → q
In p ∧ q, which scenario makes the statement false?
Both p and q are true
Either p or q is false
Both p and q are unknown
Exactly one of p and q is true
In p ∨ q, which scenario makes the statement true?
At least one of p or q is true
Both p and q are false
Exactly one of p or q is false
Neither p nor q is known
Which truth table line correctly represents p ∨ q when p=F and q=T?
p ∨ q = q
p ∨ q = p
p ∨ q = F
p ∨ q = T
Which statement best defines the exclusive or of propositions p and q?
True only when both p and q are false
True when at least one of p or q is true
True when exactly one of p or q is true
True when both p and q are true
In the truth table for p XOR q, what is the value when p is T and q is T?
T
F
Undefined
Depends on context
A class allows students who took calculus or computer science, but not both. Which logical connective models this policy?
Conjunction (∧)
Exclusive or (XOR)
Biconditional (↔)
Inclusive or (∨)
Which row makes p XOR q true?
p=F, q=F and p=T, q=T
p=T, q=T
p=T, q=F
p=F, q=F
A restaurant states, “Soup or salad comes with an entrée,” meaning not both. What is the logical interpretation?
Conjunction of soup and salad
Exclusive disjunction of soup and salad
Negation of soup with salad
Inclusive disjunction of soup and salad
What does the conditional p → q state about its truth value?
False only when p is false
False only when q is false
False when p is true and q is false
False when p and q are both true
Which assignment makes p → q false?
p=F, q=F
p=F, q=T
p=T, q=T
p=T, q=F
Let p: “Rebecca’s PC has more than 16 GB free disk space” and q: “Processor runs faster than 1 GHz.” What does p ∨ q mean?
Exactly one of disk space or speed is true
Neither disk space nor speed is true
Both disk space and processor speed are true
At least one of disk space or speed is true
Which connective matches the phrase “either A or B, but not both”?
Inclusive or (∨)
Exclusive or (⊕)
Implication (→)
Negation (¬)
Choose the correct truth table summary for XOR.
True when both inputs are equal
Always false unless both true
True when inputs are unequal
Always true unless both false
In the truth table for the conditional p → q, which row makes p → q false?
p false, q true
p true, q true
p false, q false
p true, q false
Which statement correctly expresses the conditional using p: “Maria learns discrete mathematics” and q: “Maria will find a good job”?
Maria will find a good job only if she learns discrete mathematics
Maria learns discrete mathematics if and only if she finds a good job
If Maria learns discrete mathematics, then she will find a good job
If Maria finds a good job, then she learned discrete mathematics
For the biconditional p ↔ q, when is the statement true?
When p and q have the same truth values
When q is true regardless of p
When exactly one of p or q is true
When p is true regardless of q
Which row in the biconditional truth table yields false?
p true, q true
p false, q false
p false, q true
p true, q false
Translate the biconditional using p: “You can take the flight,” and q: “You buy a ticket.”
If you can take the flight, then you will buy a ticket
You can take the flight if and only if you buy a ticket
You can take the flight if you do not buy a ticket
You buy a ticket only if you cannot take the flight
How many rows does a truth table have for n simple statements?
n² rows
2n rows
2n rows
n rows
Which best describes a truth table for a compound statement?
A chart of only the false outcomes for its components
A list of only the true rows of its components
A table of logical laws without examples
A list of all permutations of truth values for its components
Given p is false and q is true, what is p → q?
False because q is true
True by definition of conditional
True only if q is false
False because p is false
Given p is true and q is false, what is p ↔ q?
False only when both are false
True because p is true
False because their truth values differ
True because q is false
Which phrase accurately defines the biconditional connective?
q implies p, true only when both are true
p or q, true unless both are true
p implies q, true except when p and q are false
p iff q, true when p and q match
How many rows are required for the truth table of the compound [(¬r ∧ s) → t] ↔ (p ∨ ¬q), when p, q, r, s, t are primitive?
32 rows total
8 rows total
64 rows total
16 rows total
Which skill is necessary when constructing a truth table for a sentence with multiple connectives?
Selecting only true permutations
Using random truth assignments
Filling columns at given stages
Ignoring the dominant operator
For n primitive statements, what is the number of rows required in a complete truth table?
n² rows total
2n rows total
2n rows total
n rows total
In the truth table for (p ∨ ¬q) → (p ∧ q), which row yields T when p = F and q = T?
It depends on r
It is undefined
It evaluates to F
It evaluates to T
What is the dominant operator you examine to determine the compound’s truth value in a row?
The highest-precedence operator
The leftmost connective
Any operator chosen arbitrarily
Only negations in the row
Which statement best defines a Tautology for a compound proposition?
Always true for some assignments
Always false for all assignments
Always true for all assignments
Sometimes true and sometimes false
Which definition correctly describes a Contradiction?
True only when p is true
Always true for all assignments
Always false for all assignments
Neither always true nor always false
A proposition that is neither a Tautology nor a Contradiction is called what?
Contingency
Inference
Equivalence
Implication
Using the partial table, evaluate p ∧ q when p = T and q = F.
It equals p ∨ q
It is undefined
It evaluates to F
It evaluates to T
Classify the compound [p → (q → r)] → [(p → q) → (p → r)] as Tautology, Contradiction, or Contingency.
It is a Contingency
It is a Tautology
It cannot be classified
It is a Contradiction
Which statement best describes a tautology in propositional logic?
True only when antecedent is true
True for some assignments but not all
True only when consequent is false
Always true for all truth assignments
Given p, q, r are propositions, what is the meaning of p → r?
If p then r
p and r together
p or r holds
Not p implies r
In the truth table shown, why does [p → (q → r)] → [(p → q) → (p → r)] evaluate to T in the last column for all rows?
Because it is a tautology
Because p, q, r are all true
Because q → r is always false
Because p → q contradicts p → r
Which connective is used between (p → q) and (q → r) inside [(p → q) ∧ (q → r)] → (p → r)?
Conjunction ∧
Disjunction ∨
Biconditional ↔
Exclusive or ⊕
What is the final conclusion drawn from the second truth table about [(p → q) ∧ (q → r)] → (p → r)?
It is contingent
It is a contradiction
It is a tautology
It is equivalent to p ∧ q
In constructing the truth table for p → (q → r), which column must be computed before p → (q → r)?
q → r
p ↔ r
q ∨ r
p ∧ q
Which logical law is exemplified by the tautology [(p → q) ∧ (q → r)] → (p → r)?
Commutation law
De Morgan’s laws
Hypothetical syllogism
Idempotent law
If (p → q) is true and (q → r) is true, which statement must be true by the tautology highlighted?
q ↔ r must be true
p ∧ r must be true
p ↔ q must be true
p → r must be true
For the expression q ↔ (¬p ∨ ¬q), which connective relates q to (¬p ∨ ¬q)?
Disjunction ∨
Conjunction ∧
Implication →
Biconditional ↔
Which step checks whether a compound proposition is a tautology using a truth table?
Verify last column is all T
Count number of rows
Replace variables with constants
Ensure variables are distinct
In the truth table provided, what is concluded about the compound proposition q ↔ (¬p ∨ ¬q)?
It is a tautology in all rows
It is a contradiction in all rows
It is equivalent to p ∧ q
It is a contingency with T or F
According to Definition 2.10, when are two compound propositions A and B said to be logically equivalent?
When both are tautologies
When A implies B in all cases
When neither is a contradiction
When their truth values are identical
Which pair is demonstrated to be logically equivalent: ←(p ∨ q) and which other expression?
p ∨ ←q
←p ∨ ←q
←p ∧ ←q
p ∧ q
From the first demonstration table, what identity is concluded?
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p ∨ q) ≡ p ∨ q
¬(p ∨ q) ≡ p ∧ q
¬(p ∨ q) ≡ ¬p ∨ ¬q
In the second demonstration, which two expressions are shown to be logically equivalent?
p ∧ (q ∨ r) and (p ∧ q) ∨ r
p ↔ (q ∧ r) and (p ↔ q) ∧ r
p → (q ∧ r) and (p → q) ∧ (p → r)
p ∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r)
Looking at the second table, when p is F and q, r are T, what is the truth value of (p ∨ q) ∧ (p ∨ r)?
False because one disjunction is False
True because both disjunctions are True
True because p makes the conjunction True
False because both disjunctions are False
Which law is implicitly used when concluding ¬(p ∨ q) ≡ ¬p ∧ ¬q from matching truth columns?
Double Negation Law
Idempotent Law of disjunction
De Morgan’s Law for disjunction
Absorption Law for conjunction
If two propositions have identical truth tables column by column, what can be asserted?
They share at least one tautology
They contradict each other
They imply each other sometimes
They are equivalent by definition
In the table for q ↔ (¬p ∨ ¬q), what values appear in the last column across rows?
Alternating True then False
All True values appear
All False values appear
A mix of True and False
Which equivalence corresponds to the distributive property over conjunction and disjunction shown in the second demonstration?
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Which idempotent law equivalence is correct for disjunction?
p ∨ p ⇔ p
p ∨ p ⇔ ¬p
p ∨ p ⇔ T
p ∨ p ⇔ p ∨ q
What does the law of double negation state for any proposition p?
¬(¬p) ⇔ p
¬(¬p) ⇔ ¬p
¬(¬p) ⇔ p ∧ q
¬(¬p) ⇔ T
Which commutative law is valid for conjunction?
p ∧ q ⇔ q ∧ p
p ∧ q ⇔ p ∨ q
p ∧ q ⇔ ¬q ∧ p
p ∧ q ⇔ p → q
Choose the correct associative law for disjunction over three propositions p, q, r.
(p ∨ q) ∨ r ⇔ p ∨ (q ∨ r)
(p ∨ q) ∨ r ⇔ (p ∧ q) ∨ r
(p ∨ q) ∨ r ⇔ p ∧ (q ∨ r)
(p ∨ q) ∨ r ⇔ p → (q ∨ r)
Which distributive law shows disjunction distributing over conjunction?
p ∨ (q ∧ r) ⇔ (p ∧ q) ∨ (p ∧ r)
p ∨ (q ∧ r) ⇔ p ∧ (q ∨ r)
p ∨ (q ∧ r) ⇔ q ∧ (p ∨ r)
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)
Select the correct De Morgan’s law for negating a conjunction.
¬(p ∧ q) ⇔ p ∧ ¬q
¬(p ∧ q) ⇔ ¬p ∧ ¬q
¬(p ∧ q) ⇔ p ∨ q
¬(p ∧ q) ⇔ ¬p ∨ ¬q
Which inverse law simplifies a disjunction with negation?
p ∨ ¬p ⇔ T
p ∨ ¬p ⇔ F
p ∨ ¬p ⇔ ¬p
p ∨ ¬p ⇔ p
Identify the identity law for conjunction.
p ∧ T ⇔ p ∧ T
p ∧ T ⇔ p
p ∧ T ⇔ F
p ∧ T ⇔ T
Which absorption law is correct?
p ∨ (p ∧ q) ⇔ p ∨ q
p ∨ (p ∧ q) ⇔ p ∧ q
p ∨ (p ∧ q) ⇔ q
p ∨ (p ∧ q) ⇔ p
Choose the correct equivalence expressing implication using disjunction and negation.
p → q ⇔ p ∧ ¬q
p → q ⇔ ¬p ∨ q
p → q ⇔ p ∨ q
p → q ⇔ ¬(p ∨ q)
Which biconditional law rewrites p ↔ q using conjunctions and implications?
p ↔ q ≡ (p → q) ∧ (q → p)
p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)
p ↔ q ≡ (p ∨ q) ∧ (¬p ∨ ¬q)
p ↔ q ≡ (p → q) ∨ (q → p)
According to the biconditional laws, which is equivalent to ¬(p ↔ q)?
p ↔ ¬q
p ↔ q
p → q
p ∨ q
Which expression is logically equivalent to p ↔ q using disjunction of conjunctions?
(p ∧ q) ∨ (¬p ∧ ¬q)
(p → q) ∧ (q → p)
(p ∨ q) ∧ (¬p ∨ ¬q)
(p ∧ ¬q) ∨ (¬p ∧ q)
Which implication law from the table states (p → q) ∨ (p → r) ≡ p → (q ∨ r)?
Law (ix)
Law (viii)
Law (v)
Law (vi)
Identify the law: ¬(p → q) ≡ p ∧ ¬q.
Implication negation law
De Morgan law for implication
Double negation law
Commutative law of conjunction
Which equivalence simplifies (p → q) ∧ (p → r)?
p → (q ∧ r)
p → (q ∨ r)
(q ∨ r) → p
(q ∧ r) → p
In the worked example, which law justifies replacing [¬(¬p) ∨ ¬q] with (p ∨ ¬q)?
Double negation law
First De Morgan law
Commutative law of disjunction
Identity law for F
In the derivation, why does ¬p ∧ p ≡ F allow simplification to F ∨ (¬p ∧ ¬q)?
Because F absorbs under conjunction
Because implication replaces conjunction
Because disjunction distributes over conjunction
Because a contradiction yields false
Which final equivalence is concluded in the example on page 10?
¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬q
Which distribution law is used when transforming (¬p ∧ p) ∨ (¬p ∧ ¬q) into ¬p ∧ (p ∨ ¬q)?
Second distributive law
First distributive law
Idempotent law
Absorption law
Which law justifies transforming ¬[¬[(p ∨ q) ∧ r] ∨ ¬q] into ¬¬[(p ∨ q) ∧ r] ∧ ¬¬q?
Absorption law for implication
Commutative law of conjunction
Law of double negation
De Morgan’s law on disjunction
After applying De Morgan’s law and double negation to ¬[¬[(p ∨ q) ∧ r] ∨ ¬q], which equivalent form is obtained?
[(p ∨ q) ∧ r] ∧ q
(p ∨ q) ∧ (q ∧ r)
(p ∨ q) ∧ (r ∧ q)
(q ∧ r) ∧ (p ∨ q)
Which property allows rearranging (p ∨ q) ∧ (r ∧ q) to (p ∨ q) ∧ (q ∧ r)?
Domination law with T
Inverse law for q ∧ ¬q
Associative law of conjunction
Commutative law of disjunction
Which law reduces [(p ∨ q) ∧ q] ∧ r to (q ∧ r)?
Absorption law
Idempotent law
Associative law
Conditional law
From the derivation, which two statements are concluded to be logically equivalent?
¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q
¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q
Which transformation begins the proof that (p ∧ q) → (p ∨ q) is a tautology?
Absorbing q into p ∨ q
Commuting terms in a disjunction
Applying De Morgan’s law to a conjunction
Using conditional law to rewrite implication
Which step uses De Morgan’s law in the tautology proof of (p ∧ q) → (p ∨ q)?
(p ∧ q) ∨ (p ∨ q)
¬(p ∧ q) ∨ (p ∨ q)
(¬p ∨ ¬q) ∨ (p ∨ q)
(¬p ∧ ¬q) ∨ (p ∨ q)
Which laws justify rearranging (¬p ∨ p) ∨ (¬q ∨ q) in the tautology proof?
Associative and commutative laws
Idempotent and absorption laws
Distributive and conditional laws
Commutative and inverse laws
Which result is obtained by applying inverse laws to (¬p ∨ p) ∨ (¬q ∨ q)?
F ∨ F
p ∨ q
p ∧ q
T ∨ T
For a conditional p → q, which is the contrapositive?
¬q → ¬p
q → p
p → ¬q
¬p → ¬q
Given statement: p → q. Which form is logically equivalent to p → q?
q → p
¬p → ¬q
¬q → ¬p
p ∧ q
For the statement "If oxygen is a gas then gold is compound" with p: oxygen is a gas and q: gold is compound, what is the converse?
¬p → ¬q
q → p
¬q → ¬p
p → q
For p → q, which transformation yields the inverse?
p ∨ q
q → p
¬q → ¬p
¬p → ¬q
Consider "If it is raining, then the home team wins" with p: it is raining and q: home team wins. Which is the contrapositive?
q → p
¬p → ¬q
¬q → ¬p
p → q
Which statement is NOT equivalent to p → q?
¬q → ¬p
p ∧ ¬q is false
q is true or ¬p is true
q → p
Translate: "If oxygen is not a gas then Gold is not a compound" using p and q as defined.
¬q → ¬p
q → p
p → q
¬p → ¬q
Translate: "If the home team does not win, then it is not raining" using p and q as defined.
¬q → ¬p
p → q
q → p
¬p → ¬q
Definition of dual: When forming the dual s^d of a statement s with only ∨, ∧, ¬, what replacements are made?
Replace ∨ by ∧ only
Swap ∨ and ∧; swap T and F
Swap variables p and q only
Replace ¬ by ∨; replace ∧ by ¬
Find the dual of s: ¬(p ∧ q) ∨ (p ∨ q).
(p ∨ q) ∧ ¬(p ∧ q)
¬(p ∨ q) ∧ (p ∧ q)
(p ∧ q) ∨ ¬(p ∨ q)
¬p ∧ ¬q ∨ p ∨ q
Given s: (p ∧ ¬q) ∨ (r ∧ T). What is sd ?
(p ∨ q) ∧ (r ∨ T)
(p ∧ ¬q) ∨ (r ∧ F)
(p ∧ q) ∨ (r ∧ F)
(p ∨ ¬q) ∧ (r ∨ F)
Which statement best expresses the principle of duality for formulas using only ∨, ∧, and ¬?
Replacing ∨ with ∧ and swapping T with F
Replacing ∨ with ∧ and ¬ with →
Replacing ∧ with ↔ and swapping variables
Replacing ∨ with ↔ and negating every variable
Under duality, what is the dual of the conditional r → s when written as ¬r ∨ s?
¬r ∧ s
¬r ∨ ¬s
r ∧ ¬s
r ∨ ¬s
In the worked verification, which law justifies simplifying (p ∧ q) ∨ ¬p ∨ q to (p ∧ q) ∨ ¬p ∨ q with duplicated q removed?
Commutative law of ∨
Absorption law of ∨
Associative law of ∨
Idempotent law of ∨
Which step corresponds to the definition of conditional used in the solution sequence?
Replace r → s by ¬r ∨ s
Replace r → s by r ∧ ¬s
Replace r → s by ¬(r ∧ s)
Replace r → s by r ∨ ¬s
The NAND operator p ↑ q is logically equivalent to which expression?
¬(p ∨ q)
p ∧ ¬q
¬(p ∧ q)
p ↔ ¬q
When p and q are both true, what is the value of p ↑ q according to the truth table?
Both true and false
Undefined
False
True
Which description correctly characterizes NAND in terms of truth conditions?
True exactly when at least one input is false
True exactly when both inputs are false
True exactly when both inputs are true
True exactly when inputs are equal
From the given truth table, which row yields T for p ↑ q?
p=T, q=F
p=F, q=F
p=T, q=T
p=F, q=T
Using duality, what is the dual of the expression ¬p ∨ q?
p ∧ ¬q
¬(p ∧ q)
¬p ∧ q
p ∨ ¬q
Which law allows rearranging ((p ∧ q) ∨ ¬p ∨ q) as (q ∨ ¬p ∨ (p ∧ q)) in the derivation?
Commutative law of ∨
Associative law of ∨
Absorption law of ∨
Distributive law of ∨ over ∧
Which statement defines the NOR connective p ↓ q in terms of standard operators?
p ↓ q is equivalent to (¬p ∨ ¬q)
p ↓ q is equivalent to (p → ¬q)
p ↓ q is equivalent to ¬(p ∧ q)
p ↓ q is equivalent to ¬(p ∨ q)
In the truth table for p ↓ q, what is the value when p = F and q = F?
T for both false inputs
F for both false inputs
T when exactly one is false
F when exactly one is false
Peirce's arrow is another name for which logical operator?
Material implication connective
Exclusive OR connective in logic
NAND connective in Boolean algebra
NOR connective in propositional logic
Which row pattern matches the truth table of p ↓ q for inputs (T,T), (T,F), (F,T), (F,F)?
F, T, T, F
T, F, F, F
T, T, F, F
F, F, F, T
Choose the correct equivalence proven using truth tables:
¬(p ↓ q) ⇔ (¬p ↑ ¬q)
¬(p ↓ q) ⇔ (p ↑ q)
¬(p ↓ q) ⇔ (p ∨ q)
¬(p ↓ q) ⇔ (¬p ∧ ¬q)
Given p = T and q = F, what is the value of ¬(p ↓ q)?
F because p ∧ q is F
T because p ∨ q is T
F because p ↓ q is T
T because p ↓ q is F
Identify the equivalence shown by the second truth table:
¬(p ↑ q) ⇔ (p ↓ q)
¬(p ↑ q) ⇔ (p ∧ q)
¬(p ↑ q) ⇔ (¬p ↓ ¬q)
¬(p ↑ q) ⇔ (¬p ∨ ¬q)
Which statement about NOR truth values is correct?
It is true when at least one input is true
It is true only when both inputs are true
It is true only when both inputs are false
It is true exactly when inputs differ
If p and q are primitive statements, which is a valid equivalence involving NOR?
p ↓ q ⇔ (p ∧ ¬q)
p ↓ q ⇔ ¬(p → q)
p ↓ q ⇔ ¬(p ∨ q)
p ↓ q ⇔ (¬p → q)
Using the provided tables, which compound has the final column T,F,F,F from top to bottom?
p ∨ q in standard logic
p ↓ q (NOR operator)
p ⊕ q (XOR operator)
p ↑ q (NAND operator)
Let p: "ΔABC is isosceles" and q: "ΔABC is equilateral". Which English statement correctly represents p ∧ ¬q?
ΔABC is isosceles and not equilateral
ΔABC is equilateral and not isosceles
ΔABC is not isosceles and equilateral
ΔABC is neither isosceles nor equilateral
Let p: "ΔABC is isosceles" and q: "ΔABC is equilateral". Which English statement matches p → q?
If ΔABC is equilateral then it is isosceles
ΔABC is both isosceles and equilateral
If ΔABC is isosceles then it is equilateral
ΔABC is neither isosceles nor equilateral
Given p is true and q is false, what is the truth value of ¬(p → q)?
Both true and false
Cannot be determined
False
True
Given p is true and q is false, evaluate (p → q) ∨ ¬(p ↔ ¬q).
True
False
Always false
Always true
Which is logically equivalent to q → p?
¬q ∨ p
p ∨ q
p ∧ ¬q
¬p ∨ q
Given p is true and q is false, what is the truth value of (p ∧ q) → (p ∨ q)?
Depends on p only
False
Depends on q only
True
If p ∧ q is false and q is true, what is the truth value of p?
True
True only if q is false
Cannot be determined
False
If p ∨ q is false and q is false, what must be the truth value of p?
Cannot be determined
False
True only if q is true
True
Suppose p ↔ q is true and p is false. What is the truth value of q?
False
Cannot be determined
True only if p is true
True
Choose the correct truth table row for p → ¬q when p is true and q is false.
p = F, q = F, p → ¬q is False
p = T, q = F, p → ¬q is True
p = F, q = T, p → ¬q is True
p = T, q = F, p → ¬q is False
