Mathematical Reasoning and Problem Solving
Mathematical Reasoning and Problem Solving tests statements, negation, implication, converse, contrapositive and counterexamples. It is the domain where precise wording matters more than calculation.
Core rules
- A statement is a sentence that is either true or false, but not both.
- The negation of “all A are B” is “some A are not B”; the negation of “some A are B” is “no A are B”.
- For the implication p → q: converse is q → p; contrapositive is ¬q → ¬p; inverse is ¬p → ¬q. The contrapositive is logically equivalent to the original.
- A single counterexample disproves a universal statement.
- “And” requires both conditions; “or” requires at least one.
Worked example
Consider “If a number is divisible by 4, then it is even.” This is true. Its converse, “If a number is even, then it is divisible by 4,” is false: 6 is even but not divisible by 4, so 6 is a counterexample. The contrapositive, “If a number is not even, then it is not divisible by 4,” is true and equivalent to the original.
Traps
The converse is not the contrapositive. Negating “at least one” gives “none”, not “at least one is not”. A statement with a free variable, such as “x + 2 = 5”, is not a statement until x is specified. Do not prove a universal claim by checking only examples; one counterexample is enough to refute it.