Logical Reasoning · Form 4
Logical Reasoning: Worked Examples (Easier)
This set drills the single core skills of logical reasoning: negating a statement, judging the truth of an 'and' statement, and testing a quantifier. Ideal for students meeting the topic for the first time.
Worked example 1
Write the negation of the statement 'p: 9 is a factor of 45', and state whether the negation is true or false.
- Read the statement p: '9 is a factor of 45.'
- Check p: 9 × 5 = 45, so 9 divides 45 exactly; p is true.
- Form the negation ~p by inserting 'is not': '9 is not a factor of 45.'
- Since p is true, its negation ~p must be false.
Worked example 2
Determine whether the compound statement '7 is a prime number and 12 is a multiple of 5' is true or false.
- Split into two simple statements joined by 'and'.
- p: '7 is a prime number' 7 has only factors 1 and 7, so p is true.
- q: '12 is a multiple of 5' multiples of 5 are 5, 10, 15, ...; 12 is not among them, so q is false.
- An 'and' statement is true only when both parts are true; here q is false, so the whole statement is false.
Worked example 3
State the quantifier used in 'All multiples of 4 are even numbers', and determine whether the statement is true or false.
- Identify the quantifier word: 'All' a universal quantifier.
- Write a general multiple of 4 as 4 × n, where n is a whole number.
- 4 × n = 2 × (2n), which is 2 times a whole number, so it is always even.
- No multiple of 4 can be odd, so the statement holds for every case and is true.
Worked example 4
Write the negation of the statement 'q: 8 is a multiple of 3' and state whether the negation is true or false.
- The original statement says 8 is a multiple of 3.
- The negation inserts 'not': '8 is not a multiple of 3'.
- Check: 3×2 = 6 and 3×3 = 9, so 8 is not a multiple of 3.
- Therefore the negation is true.
Worked example 5
Determine whether the statement '5 is an even number or 5 is a prime number' is true or false.
- The statement is joined by 'or' (a disjunction); it is true if at least one part is true.
- '5 is an even number' is false.
- '5 is a prime number' is true.
- Since at least one part is true, the whole 'or' statement is true.
Worked example 6
State the quantifier used in 'Some prime numbers are even', and state whether the statement is true or false.
- The quantifier is the word showing quantity: here it is 'Some'.
- 'Some' means at least one exists.
- The prime number 2 is even, so at least one prime number is even.
- Therefore the statement is true.
Worked example 7
Write the negation of the statement 'r: A rectangle has four right angles', and state whether the negation is true or false.
- The negation of r is: ~r: 'A rectangle does not have four right angles.'
- Check: by definition, every rectangle has four right angles, so r is true.
- Since r is true, its negation ~r must be false.
Worked example 8
State the quantifier used in the statement 'Some triangles have a right angle', and determine whether the statement is true or false.
- The quantifier used is 'some'.
- A right-angled triangle (for example one with angles 90°, 60° and 30°) does have a right angle.
- Since at least one triangle satisfies the condition, the statement is true.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know if a sentence counts as a "statement" in this chapter?
A statement must be a sentence that is definitely true or definitely false, not both and not neither. Questions, commands, and opinions ("this is a nice day") are not statements because their truth value can't be fixed, this distinction is tested directly in exam questions.
What's the easiest way to write the negation of a statement correctly?
Add "not" in the right place, or rewrite the statement so its truth value flips completely. A common mistake is changing the meaning instead of just the truth value, always check that the negation is true exactly when the original statement is false.
Why do I keep confusing "and" and "or" statements?
A conjunction ("and") is only true when both parts are true, while a disjunction ("or") is true when at least one part is true. Practising with simple true/false examples for each part helps you build the habit of checking both parts carefully before deciding.