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Mathematical reasoning
Mathematics
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Overview
Mathematical reasoning evaluates statements and arguments using logic: connectives, implications, quantifiers and methods of proof.
Key points
- A statement is either true or false, not both.
- "p and q" is true only when both are true; "p or q" is true when at least one is true.
- For "if p then q", the contrapositive "if not q then not p" is logically equivalent.
- The converse "if q then p" is not necessarily true.
- A single counterexample disproves a universal statement.
- Mathematical induction proves statements for all natural numbers.
Explanation
- Negation of "all" is "some are not".
- Patterns in numbers suggest conjectures that need proof.
- Children’s reasoning develops from concrete examples to general rules.
- Proof by contradiction assumes the opposite.
- Reasoning tasks build problem-solving skills.
Step by step
- Identify the concept, rule or fact asked.
- Recall the definition, formula, example or year.
- Apply it and eliminate mismatched options.
Examples
- Statement: "If a number is divisible by 4, it is even." Contrapositive: "If a number is not even, it is not divisible by 4." (true)
- Converse "If a number is even, it is divisible by 4" → false (6 is a counterexample).
- Negation of "All birds can fly" → "Some birds cannot fly".
Traps and advanced points
- Trap: Treating converse as equivalent.
- Trap: Wrong negation of "all".
- Trap: Using examples as proof.
Quick revision
- Contrapositive equivalent.
- Counterexample disproves.
- Induction.
Practice MCQs
A statement in logic is —
- a question
- a command
- either true or false
- always true
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Answer: (c) either true or false. Statements have a definite truth value.
"p and q" is true when —
- at least one is true
- both are true
- both are false
- p is false
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Answer: (b) both are true. Conjunction needs both true.
"p or q" is false when —
- one is true
- both are true
- p is true
- both are false
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Answer: (d) both are false. Disjunction is false only if both are false.
The contrapositive of "if p then q" is —
- p and q
- if q then p
- if not p then not q
- if not q then not p
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Answer: (d) if not q then not p. Contrapositive is equivalent.
The converse of "if p then q" is —
- if q then p
- if not q then not p
- not p
- p or q
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Answer: (a) if q then p. Swap hypothesis and conclusion.
The negation of "All birds can fly" is —
- No bird can fly
- Some birds cannot fly
- All birds cannot fly
- Birds fly
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Answer: (b) Some birds cannot fly. Negation of "all" is "some … not".
A single counterexample can —
- prove a universal statement
- disprove a universal statement
- prove an existential statement false
- do nothing
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Answer: (b) disprove a universal statement. One exception disproves "all".
"If a number is divisible by 4, it is even" — the converse is —
- a contrapositive
- true
- false (6 is a counterexample)
- equivalent
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Answer: (c) false (6 is a counterexample). 6 is even but not divisible by 4.
Mathematical induction proves statements for —
- negative numbers only
- all natural numbers
- one number only
- fractions only
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Answer: (b) all natural numbers. Base case + inductive step.
Proof by contradiction begins by —
- listing examples
- guessing
- assuming the opposite of what is to be proved
- drawing a graph
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Answer: (c) assuming the opposite of what is to be proved. A contradiction shows the assumption false.
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