Logical reasoning in SPM Mathematics is about stating things precisely and testing whether a conclusion follows. This page shows how the topics connect and where to begin.
The chapter sits inside SPM Mathematics, and its habits carry into proof-style questions elsewhere in the syllabus.
What does logical reasoning cover?
Four skills make up the chapter, and each builds on the last.
- Deciding whether a sentence is a statement with a truth value
- Negating statements, including ones with “all” and “some”
- Writing implications, converses and other forms of “if … then”
- Testing whether an argument is valid, using counterexamples
Work through them in this order.
How do the parts connect?
Every later skill uses the one before it. You cannot negate a sentence until you know it is a statement. You cannot test a converse until you can write the implication clearly.
- Distinguishing statements from non-statements gives you the basic unit.
- Negating statements with quantifiers teaches you to flip a statement correctly.
- Writing implications and their converse covers the “if … then” form.
- Testing deductive arguments with counterexamples uses all three to judge an argument.
One statement, three ways
Take the statement “All the students in Form 5 Bestari wear a school tie.” It is true or false, so it is a statement.
Its negation is “Some students in Form 5 Bestari do not wear a school tie.” The word “all” changes to “some”, and “wear” becomes “do not wear”.
Written as an implication, it becomes “If a person is in Form 5 Bestari, then the person wears a school tie.” Its converse, “If a person wears a school tie, then the person is in Form 5 Bestari”, is a different claim. A student from another class who wears a tie is a counterexample to the converse.
One sentence gave us a negation, an implication and a converse, so each skill needs the one before it.
Who should start where?
If you are unsure what counts as a statement, start at lesson 1. If you can do that but your negations come out wrong, start at lesson 2. If your basics are solid, go straight to the practice set and use the lessons to repair what you miss.
There is also a page on proving or disproving an everyday mathematical claim, which shows the reasoning applied to a claim you might hear outside class.
A mistake log helps here, because the same wording slips come back. For lessons with a teacher, see online one-to-one Mathematics tuition.