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Logical Reasoning · Form 4

Logical Reasoning: Worked Examples (KBAT)

This set hides the technique inside everyday situations, a number pattern, a school badge rule, and a cafe poster, so students must decide between inductive reasoning, the converse trap, and disproving an 'all' claim with one counterexample.

Written by the spmmath.com.my editorial team.

Worked example 1

While revising, Mei notices a pattern: 1 × 8 + 1 = 9, 12 × 8 + 2 = 98, 123 × 8 + 3 = 987, 1234 × 8 + 4 = 9876. Using inductive reasoning, write her conjecture for the next line and use it to state the value of 12345 × 8 + 5.

  1. Read each answer: 9, 98, 987, 9876, the digits count down from 9 and one more digit appears each line.
  2. Inductive reasoning: from these repeated cases, conjecture that 12345 × 8 + 5 continues the pattern as 98765.
  3. Check by direct calculation: 12345 × 8 = 98760.
  4. Add the 5: 98760 + 5 = 98765, which matches the conjecture.

Worked example 2

A school rule states: 'If a student is a prefect, then the student wears a blue badge.' At assembly a teacher sees that Chong is wearing a blue badge.

Chong claims this proves he is a prefect. Explain, using implication and its converse, whether Chong's claim must be correct.

  1. Write the rule as an implication p → q, where p is 'is a prefect' and q is 'wears a blue badge'.
  2. The teacher observes q is true (Chong wears a blue badge); Chong wants to conclude p is true.
  3. Concluding p from q uses the converse q → p ('if wears a blue badge, then is a prefect'), which the rule does not give.
  4. The converse need not be true, other students, e.g. librarians, might also wear blue badges, so q being true does not force p to be true.

Worked example 3

A cafe poster claims: 'All our drinks contain no added sugar.' A customer tests the mango smoothie and finds it does contain added sugar.

Write the negation of the poster's claim, and explain whether one such drink is enough to show the claim is false.

  1. The claim uses the universal quantifier 'all': every drink has no added sugar.
  2. The negation of 'all ... have no added sugar' is 'at least one drink contains added sugar' (a 'some' statement).
  3. To disprove an 'all' statement you only need one counterexample.
  4. The mango smoothie contains added sugar, so it is that counterexample and the poster's claim is false.

Worked example 4

Study the pattern: 1×9+2 = 11, 12×9+3 = 111, 123×9+4 = 1111. Make a conjecture about the pattern and use it to write the value of 12345×9+6.

  1. Line 1 uses one digit and gives 11 (two 1s); line 2 uses two digits and gives 111 (three 1s); line 3 uses three digits and gives 1111 (four 1s).
  2. Conjecture: when 123…n is multiplied by 9 and (n+1) is added, the answer is a string of (n+1) ones.
  3. For 12345×9+6 the number uses five digits (n = 5), so the answer has 5 + 1 = 6 ones.
  4. Check: 12345×9 = 111105, and 111105 + 6 = 111111.

Worked example 5

In a chess club it is known that 'All members of the chess club can play chess.' Rajan cannot play chess.

Using logical reasoning, determine whether Rajan is a member of the chess club and name the type of reasoning used.

  1. The rule 'if a person is a member, then that person can play chess' is an implication.
  2. Its contrapositive is 'if a person cannot play chess, then that person is not a member', which is true whenever the original is true.
  3. Rajan cannot play chess, so by the contrapositive he is not a member.
  4. This is valid deductive reasoning: a general rule applied to a specific case.

Worked example 6

A student writes: '3, 5 and 7 are all prime, so every odd number is prime.' Explain why this reasoning is not valid, and give the smallest odd number greater than 1 that shows it is wrong.

  1. Checking a few cases (3, 5, 7) is inductive reasoning; a few examples cannot prove a claim about every odd number.
  2. To disprove 'every odd number is prime' we only need one odd number that is not prime, a counterexample.
  3. Test odd numbers beyond 7: 9 is odd, but 9 = 3×3, so 9 is not prime.
  4. Therefore the claim is false, and 9 is the smallest odd number greater than 1 that is not prime.

Worked example 7

A theme park ride has the rule: 'A rider must be at least 12 years old and at least 140 cm tall.' Amir is 13 years old but only 135 cm tall.

Using logical reasoning, determine whether Amir may go on the ride, and explain which type of compound statement applies.

  1. The rule combines two conditions with 'and', so it is a conjunction: p: 'rider is at least 12 years old' and q: 'rider is at least 140 cm tall'.
  2. Check p: Amir is 13 years old, so p is true.
  3. Check q: Amir is 135 cm tall, which is less than 140 cm, so q is false.
  4. A conjunction 'p and q' is true only when both p and q are true; here p is true but q is false, so 'p and q' is false.

Worked example 8

While practising mental arithmetic, Hafiz notices this pattern: 11² = 121, 111² = 12321, 1111² = 1234321. Using inductive reasoning, state his conjecture for the next line and use it to write the value of 11111².

  1. Each line squares a string of 1's, and the digits of the answer count up from 1 to the number of 1's, then back down to 1 (e.g. 1111² = 1234321 counts 1-2-3-4-3-2-1).
  2. Conjecture: 11111² follows the same pattern, counting up to 5 and back down: 123454321.
  3. Applying the conjecture: 11111² = 123454321.

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

How do I tell deductive reasoning apart from inductive reasoning in an exam answer?

Deductive reasoning applies a general rule to reach a specific, certain conclusion, while inductive reasoning generalises from specific examples or patterns and may not always be true. State which type you're using explicitly, since examiners check that your reasoning method matches what the question asks for.

What makes a good counter-example in these questions?

A counter-example is a single specific case where a general statement is false, it must be concrete (an actual number or object), not another general claim. One valid counter-example is enough to disprove a statement, so choose the simplest case that clearly fails.

Why do KBAT logical reasoning questions feel harder than the formula-based chapters?

Because they test reading and reasoning rather than calculation, you must interpret real statements, decide their truth value, and justify conclusions in words. Practise explaining your reasoning step by step in full sentences, since marks are awarded for the logic shown, not just a final yes/no.

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