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Mathematical Reasoning · Quick Reference
Logical Reasoning Cheat Sheet
In one line: logical reasoning is about drawing valid conclusions from given statements — separating what must be true from what merely might be true.
Key Ideas
1Statements and Conclusions. Given one or more statements assumed to be true, a conclusion follows logically only if it must be true in every possible scenario consistent with those statements.
2Syllogisms. Arguments built from two premises and a conclusion involving categories, such as 'All A are B. All B are C. Therefore, all A are C.'
3Valid vs. Invalid Conclusions. A conclusion is valid only if it necessarily follows; a conclusion that is merely likely or possible (but not guaranteed) is invalid, even if it sounds reasonable.
4Blood Relation Puzzles. Problems that require tracing family relationships step by step from a set of given clues to answer a specific question.
5Direction Sense Problems. Problems involving movement in compass directions (North, South, East, West), often solved by sketching a simple path diagram.
Worked Examples
Statements: All roses are flowers. All flowers need water. Conclusion: All roses need water. Is this valid?
Yes, the conclusion is valid
Statements: Some cats are black. Some black things are cars. Conclusion: Some cats are cars. Is this valid?
No, the conclusion does not necessarily follow
A man walks 5km North, then 3km East, then 5km South. How far is he from his starting point, and in what direction?
He is 3km East of his starting point
Practice Yourself
Statements: All dogs are mammals. All mammals are animals. Conclusion: All dogs are animals. Valid?
Yes, valid
Statements: Some birds can fly. Penguins are birds. Conclusion: Penguins can fly. Valid?
No, invalid — 'some' doesn't mean 'all'
A person walks 4km East then 4km West. How far from the start are they?
0km, back at the start
If A is B's father and B is C's father, what is A to C?
A is C's grandfather