These eight questions cover the four skills in the logical reasoning chapter. They ramp up in difficulty, so later questions combine earlier ideas.
Write your answers in full sentences before opening each answer. The lessons are linked at the end for any question you miss.
Questions
Question 1. Which of these is a statement? Give its truth value.
(a) 14 + 3 = 17 (b) Is 14 an even number? (c) x − 2 = 8
Answer
(a) is a statement and it is true, because 14 + 3 = 17. (b) is a question, so it is a non-statement. (c) depends on x, so it is an open sentence and a non-statement.
Question 2. Write the negation of “All rhombuses have four equal sides.” State whether the original or the negation is true.
Answer
Negation: “Some rhombuses do not have four equal sides.” A rhombus has four equal sides by definition, so the original is true and the negation is false.
Question 3. Write the negation of “Some prime numbers are even.” Which is true?
Answer
Negation: “No prime numbers are even.” The original is true because 2 is prime and even, so the negation is false.
Question 4. Write the statement “If a number is divisible by 10, then it is divisible by 5” and its converse. Is each one true?
Answer
The original is true, because every multiple of 10 can be written as 5 × 2k, which is a multiple of 5.
The converse is “If a number is divisible by 5, then it is divisible by 10.” It is false, because 15 is divisible by 5 but not by 10.
Question 5. “x + 4 = 9 if and only if x = 5.” Write the two implications this statement contains.
Answer
Implication 1: If x + 4 = 9, then x = 5. Implication 2: If x = 5, then x + 4 = 9.
Both are true. Subtract 4 to get the first, and add 4 to get the second, so the “if and only if” statement is true.
Question 6. Premise 1: All multiples of 8 are even.
Premise 2: 56 is a multiple of 8. State the conclusion and the type of argument.
Answer
Conclusion: 56 is even. This is a deductive argument, because the conclusion follows from the premises with certainty. Check: 56 = 8 × 7, and 8 × 7 is even.
Question 7. Premise 1: All cats have tails. Premise 2: Bobo has a tail.
Conclusion: Bobo is a cat. Is the argument valid? Give a reason.
Answer
It is not valid. The first premise says cats have tails, but it does not say that only cats have tails. Bobo could be a dog with a tail, and that counterexample makes both premises true while the conclusion is false.
Question 8. Sim asks three friends in her class if they like Mathematics, and all three say yes. She concludes, “All Form 5 students like Mathematics.” State the type of argument and comment on its strength.
Answer
This is an inductive argument, because it goes from a few cases to a general claim. It is weak, since three friends are a tiny sample of all Form 5 students, and one student who dislikes Mathematics would contradict it.
If you got these wrong
- Questions 1 and 2: revisit statements and non-statements and negating statements with quantifiers.
- Questions 4 and 5: read writing implications and their converse.
- Questions 6 to 8: read testing deductive arguments with counterexamples.
The timed original practice session builder is a way to repeat this kind of set under time. Log each wrong answer in the mistake log, and for a teacher who can ask you to explain your choices, see online one-to-one Mathematics tuition.