Completing Statements MCQs for Sub-Topics of Topic 11: Mathematical Reasoning

Question 1. A mathematical statement, or proposition, is a declarative sentence that____

(A) is grammatically correct.

(B) expresses an opinion.

(C) is either true or false, but not both.

(D) asks a question.

Question 2. The sentence "$x > 7$" is considered an open sentence because____

(A) it is a question.

(B) its truth value depends on the value of $x$.

(C) it contains a mathematical symbol.

(D) it is a command.

Question 3. The truth value of the statement "The Sun revolves around the Earth" is____

(A) True.

(B) False.

(C) Undetermined.

(D) Subjective.

Question 4. A sentence cannot be a mathematical statement if it is a(n)____

(A) declarative sentence.

(B) sentence with a definite truth value.

(C) imperative sentence.

(D) assertion of fact.

Question 5. For a statement to be considered true, it must____

(A) be widely believed.

(B) be consistent with mathematical definitions and axioms, or observed reality if empirical.

(C) have a subjective truth value of true for the person making the statement.

(D) contain only numbers.

Question 6. The statement "$5 + 8 = 13$" is a proposition with a truth value of____

(A) False.

(B) True.

(C) Dependent on context.

(D) Undefined.

Question 7. The sentence "Go to the market!" is not a proposition because it is a(n)____

(A) declarative sentence.

(B) opinion.

(C) imperative sentence.

(D) open sentence.

Question 8. If a sentence's truth value cannot be determined objectively, it is typically classified as____

(A) a true statement.

(B) a false statement.

(C) not a proposition.

(D) a subjective statement.

Question 9. A sentence like "This statement is false" is problematic in logic because it leads to a____

(A) tautology.

(B) contradiction.

(C) paradox.

(D) simple statement.

Question 10. In mathematics, statements are foundational because they are the building blocks for forming____

(A) opinions.

(B) questions.

(C) proofs and arguments.

(D) commands.

Question 1. The negation of a statement $p$, denoted by $\neg p$ or $\sim p$, is a statement that is true precisely when $p$ is____

(A) true.

(B) false.

(C) a compound statement.

(D) undefined.

Question 2. To write the negation of a simple statement like "The book is heavy", you would typically add the word "not" or an equivalent phrase, resulting in____

(A) Is the book heavy?

(B) The book is light.

(C) The book is not heavy.

(D) The book and the table are heavy.

Question 3. A compound statement is formed by combining two or more simple statements using____

(A) variables.

(B) truth values.

(C) logical connectives.

(D) predicates.

Question 4. If $p$ is the statement "It is hot" and $q$ is the statement "It is sunny", the compound statement "It is hot and sunny" is symbolised as____

(A) $p \lor q$.

(B) $\neg p \land q$.

(C) $p \land q$.

(D) $p \implies q$.

Question 5. The negation of the statement "Some animals can talk" is logically equivalent to the statement____

(A) Some animals cannot talk.

(B) All animals can talk.

(C) No animal can talk.

(D) All animals cannot talk.

Question 6. The symbol $\neg$ or $\sim$ is used to represent the logical operation called____

(A) conjunction.

(B) disjunction.

(C) negation.

(D) implication.

Question 7. If a simple statement $p$ is false, then the truth value of its negation $\neg p$ is____

(A) True.

(B) False.

(C) Undetermined.

(D) The same as $p$.

Question 8. The statement "The number is an integer or a fraction" is a compound statement formed using the connective____

(A) and.

(B) not.

(C) or.

(D) if____then.

Question 9. The negation of the negation of a statement $p$, denoted $\neg(\neg p)$, is logically equivalent to____

(A) $\neg p$.

(B) $p$.

(C) a contradiction.

(D) a tautology.

Question 10. Compound statements allow us to express more complex ideas by specifying relationships between____

(A) mathematical symbols.

(B) truth values.

(C) simple propositions.

(D) variables.

Question 1. The conjunction of two statements $p$ and $q$, written as $p \land q$, is true only when____

(A) $p$ is true.

(B) $q$ is true.

(C) both $p$ and $q$ are true.

(D) at least one of $p$ or $q$ is true.

Question 2. The disjunction of two statements $p$ and $q$, written as $p \lor q$ (inclusive OR), is false only when____

(A) $p$ is false.

(B) $q$ is false.

(C) both $p$ and $q$ are true.

(D) both $p$ and $q$ are false.

Question 3. If statement $p$ is true and statement $q$ is false, the truth value of $p \land q$ is____

(A) True.

(B) False.

(C) Cannot be determined.

(D) Depends on the statements $p$ and $q$.

Question 4. If statement $p$ is false and statement $q$ is true, the truth value of $p \lor q$ (inclusive OR) is____

(A) True.

(B) False.

(C) Cannot be determined.

(D) The same as $p \land q$'s truth value in this case.

Question 5. A truth table is a systematic way to determine the truth value of a compound statement for all possible combinations of truth values of its____

(A) logical connectives.

(B) variables.

(C) simple statements.

(D) symbols.

Question 6. If a compound statement involves $n$ simple propositions, its truth table will have $2^n$____

(A) columns.

(B) rows.

(C) logical connectives.

(D) truth values in the final column.

Question 7. The truth table column for $\neg p$ has the opposite truth value to the column for $p$ in____

(A) some rows.

(B) all rows.

(C) no rows.

(D) only the rows where $p$ is true.

Question 8. The logical connective corresponding to the English word "AND" is symbolised by____

(A) $\lor$.

(B) $\neg$.

(C) $\implies$.

(D) $\land$.

Question 9. In a truth table, the leftmost columns typically represent the truth values of the____

(A) compound statement.

(B) logical connectives.

(C) simple propositions.

(D) negation.

Question 10. The exclusive OR ($p \oplus q$) is true when $p$ and $q$ have____

(A) the same truth value.

(B) different truth values.

(C) both true truth values.

(D) both false truth values.

Question 1. A conditional statement of the form "If $p$, then $q$" is symbolised as____

(A) $p \land q$.

(B) $p \lor q$.

(C) $p \implies q$.

(D) $p \iff q$.

Question 2. The only case where a conditional statement $p \implies q$ is false is when____

(A) $p$ is true and $q$ is true.

(B) $p$ is false and $q$ is true.

(C) $p$ is true and $q$ is false.

(D) $p$ is false and $q$ is false.

Question 3. In the conditional statement "$p \implies q$", $p$ is called the antecedent or hypothesis, and $q$ is called the____

(A) converse.

(B) inverse.

(C) contrapositive.

(D) consequent or conclusion.

Question 4. The biconditional statement "$p$ if and only if $q$" is symbolised as____

(A) $p \implies q$.

(B) $p \land q$.

(C) $p \lor q$.

(D) $p \iff q$.

Question 5. A biconditional statement $p \iff q$ is true when $p$ and $q$ have____

(A) different truth values.

(B) the same truth value.

(C) both true truth values.

(D) both false truth values.

Question 6. The phrase "$p$ is a sufficient condition for $q$" is logically equivalent to the conditional statement____

(A) $q \implies p$.

(B) $p \implies q$.

(C) $p \land q$.

(D) $p \lor q$.

Question 7. The phrase "$q$ is a necessary condition for $p$" is logically equivalent to the conditional statement____

(A) $p \implies q$.

(B) $q \implies p$.

(C) $p \land q$.

(D) $p \lor q$.

Question 8. When the antecedent of a conditional statement is false, the conditional statement itself is always considered____

(A) True.

(B) False.

(C) Undetermined.

(D) Dependent on the consequent.

Question 9. The biconditional statement $p \iff q$ is logically equivalent to the conjunction of the two conditional statements $p \implies q$ and____

(A) $\neg p \implies \neg q$.

(B) $\neg q \implies \neg p$.

(C) $q \implies p$.

(D) $p \land q$.

Question 10. Understanding conditional statements is crucial in mathematics because they form the basis for stating theorems and performing____

(A) definitions.

(B) examples.

(C) proofs.

(D) negations.

Question 1. Given a conditional statement $p \implies q$, its converse is formed by swapping the antecedent and consequent, resulting in____

(A) $\neg p \implies \neg q$.

(B) $q \implies p$.

(C) $\neg q \implies \neg p$.

(D) $p \land \neg q$.

Question 2. The inverse of a conditional statement $p \implies q$ is formed by negating both the antecedent and the consequent, resulting in____

(A) $\neg p \implies \neg q$.

(B) $q \implies p$.

(C) $\neg q \implies \neg p$.

(D) $p \land \neg q$.

Question 3. The contrapositive of a conditional statement $p \implies q$ is formed by swapping and negating both the antecedent and the consequent, resulting in____

(A) $\neg p \implies \neg q$.

(B) $q \implies p$.

(C) $\neg q \implies \neg p$.

(D) $p \land \neg q$.

Question 4. A conditional statement $p \implies q$ is logically equivalent to its____

(A) converse.

(B) inverse.

(C) contrapositive.

(D) negation.

Question 5. The converse of a conditional statement $p \implies q$ is logically equivalent to its____

(A) inverse ($\neg p \implies \neg q$).

(B) contrapositive ($\neg q \implies \neg p$).

(C) negation ($p \land \neg q$).

(D) original statement ($p \implies q$).

Question 6. If a conditional statement is true, its converse is not necessarily true, but its contrapositive is always____

(A) false.

(B) true.

(C) undetermined.

(D) logically equivalent to the converse.

Question 7. Consider the statement "If a number is divisible by 6, then it is divisible by 3". Its inverse is "If a number is not divisible by 6, then it is not divisible by 3". This inverse statement is____

(A) True.

(B) False.

(C) Logically equivalent to the original statement.

(D) The contrapositive of the original statement.

Question 8. The relationship between a conditional statement and its contrapositive is often used in proof techniques because they share the same____

(A) meaning.

(B) truth value.

(C) simple statements.

(D) logical connectives.

Question 9. The negation of the conditional statement $p \implies q$ is logically equivalent to____

(A) $\neg p \implies \neg q$.

(B) $q \implies p$.

(C) $p \land \neg q$.

(D) $\neg p \lor q$.

Question 10. If the converse of a statement is false, then its inverse must also be____

(A) true.

(B) false.

(C) logically equivalent to the original statement.

(D) logically equivalent to the contrapositive.

Question 1. The universal quantifier, symbolised by $\forall$, is used to indicate that a statement holds for____

(A) some elements in the domain.

(B) at least one element in the domain.

(C) exactly one element in the domain.

(D) every element in the domain.

Question 2. The existential quantifier, symbolised by $\exists$, is used to indicate that a statement holds for____

(A) every element in the domain.

(B) some element(s) in the domain.

(C) no element in the domain.

(D) exactly one element in the domain.

Question 3. The statement "All dogs can bark" can be written using a universal quantifier as $\forall x, P(x) \implies Q(x)$, where $P(x)$ is "$x$ is a dog" and $Q(x)$ is____

(A) $x$ is not a dog.

(B) $x$ can bark.

(C) $x$ cannot bark.

(D) $x$ is a dog and $x$ can bark.

Question 4. The statement "There exists a student who scored above 90%" can be written using an existential quantifier as $\exists x, P(x) \land Q(x)$, where $P(x)$ is "$x$ is a student" and $Q(x)$ is____

(A) $x$ is not a student.

(B) $x$ scored above 90%.

(C) $x$ did not score above 90%.

(D) $x$ is a student $\implies x$ scored above 90%.

Question 5. The negation of the statement "All birds can fly" is equivalent to the statement____

(A) All birds cannot fly.

(B) No birds can fly.

(C) Some birds can fly.

(D) Some birds cannot fly.

Question 6. The negation of the statement "Some cats are black" is equivalent to the statement____

(A) Some cats are not black.

(B) All cats are black.

(C) No cat is black.

(D) All cats are not black.

Question 7. The statement "There is no integer $x$ such that $x^2 = -1$" is logically equivalent to____

(A) $\exists x, x^2 = -1$.

(B) $\exists x, x^2 \neq -1$.

(C) $\forall x, x^2 = -1$.

(D) $\forall x, x^2 \neq -1$.

Question 8. The negation of $\forall x, P(x)$ is logically equivalent to____

(A) $\forall x, \neg P(x)$.

(B) $\exists x, P(x)$.

(C) $\exists x, \neg P(x)$.

(D) $\neg (\exists x, P(x))$.

Question 9. The statement "Every student likes mathematics" is a statement involving the____

(A) existential quantifier.

(B) universal quantifier.

(C) negation.

(D) conjunction.

Question 10. Quantifiers are used in mathematical statements to specify the extent to which a property or relation applies over a given____

(A) predicate.

(B) variable.

(C) connective.

(D) domain or universe of discourse.

Question 1. A tautology is a compound statement that is always____

(A) true.

(B) false.

(C) logically equivalent to its negation.

(D) a contradiction.

Question 2. A contradiction, also called a fallacy, is a compound statement that is always____

(A) true.

(B) false.

(C) a tautology.

(D) a contingency.

Question 3. A contingency is a compound statement that is neither a tautology nor a contradiction, meaning its truth value____

(A) is always true.

(B) is always false.

(C) depends on the truth values of its simple statements.

(D) is always undetermined.

Question 4. Two statements $P$ and $Q$ are logically equivalent if they have the same truth value in all possible assignments of truth values to their simple propositions. This is equivalent to saying that the biconditional statement $P \iff Q$ is a(n)____

(A) contradiction.

(B) contingency.

(C) tautology.

(D) invalid argument.

Question 5. Constructing a truth table for a complex compound statement systematically lists all possible truth value combinations for simple propositions and calculates the truth value of the compound statement for each____

(A) column.

(B) row.

(C) connective.

(D) statement.

Question 6. The statement $p \land (q \lor \neg q)$ is logically equivalent to $p$ because $q \lor \neg q$ is a tautology, and $p \land \text{True}$ is equivalent to____

(A) True.

(B) False.

(C) $p$.

(D) $\neg p$.

Question 7. If the final column of a truth table for a compound statement contains a mix of 'True' and 'False' values, the statement is classified as a____

(A) tautology.

(B) contradiction.

(C) contingency.

(D) logical equivalence.

Question 8. De Morgan's law $\neg (p \land q) \equiv \neg p \lor \neg q$ shows a logical equivalence between the negation of a conjunction and the disjunction of the____

(A) original statements.

(B) negations of the statements.

(C) converse statements.

(D) contrapositive statements.

Question 9. The statement $(p \land \neg p) \implies q$ is always true, regardless of $q$'s truth value, because the antecedent $p \land \neg p$ is a(n)____

(A) tautology.

(B) contradiction.

(C) contingency.

(D) simple statement.

Question 10. Logical equivalence allows us to replace a statement with another statement that has the same____

(A) meaning.

(B) structure.

(C) truth table column.

(D) number of connectives.

Question 1. Validating a mathematical statement involves establishing its truth through a rigorous logical argument called a(n)____

(A) example.

(B) opinion.

(C) proof.

(D) definition.

Question 2. A direct proof of a conditional statement $p \implies q$ begins by assuming the truth of the antecedent $p$ and proceeds by using definitions, axioms, and logical rules to show that the consequent $q$ must also be____

(A) false.

(B) undetermined.

(C) true.

(D) a contradiction.

Question 3. Proof by contrapositive of $p \implies q$ involves proving the logically equivalent statement $\neg q \implies \neg p$, which means you start by assuming $\neg q$ is true and deduce that $\neg p$ must be____

(A) false.

(B) true.

(C) a contradiction.

(D) the antecedent.

Question 4. In a proof by contradiction for a statement $P$, you start by assuming $\neg P$ is true and then derive a contradiction. This leads to the conclusion that the initial assumption ($\neg P$) was false, and therefore $P$ must be____

(A) false.

(B) undetermined.

(C) a contradiction.

(D) true.

Question 5. An argument is considered valid if the truth of the premises guarantees the truth of the conclusion. This means it's impossible for all premises to be true while the conclusion is____

(A) true.

(B) false.

(C) a tautology.

(D) a contradiction.

Question 6. Modus Ponens is a valid argument form that states if you have a conditional statement $p \implies q$ and the premise $p$, you can logically conclude____

(A) $\neg p$.

(B) $\neg q$.

(C) $q$.

(D) $p \land q$.

Question 7. Modus Tollens is a valid argument form that states if you have a conditional statement $p \implies q$ and the premise $\neg q$, you can logically conclude____

(A) $p$.

(B) $q$.

(C) $\neg p$.

(D) $p \land \neg q$.

Question 8. A sound argument is a valid argument where, in addition to the conclusion logically following from the premises, the premises themselves are also____

(A) false.

(B) irrelevant.

(C) true.

(D) contradictory.

Question 9. To demonstrate that a universally quantified statement like "All odd numbers are prime" is false, you only need to find a single counterexample, such as the number____

(A) 2.

(B) 3.

(C) 9.

(D) 5.

Question 10. Proof by contradiction is particularly useful for proving statements about non-existence, such as proving that there is no largest prime number or that $\sqrt{2}$ is____

(A) rational.

(B) irrational.

(C) an integer.

(D) a perfect square.