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📘 Lesson 1 of 4 · Mathematical Reasoning

🧩 Logical Reasoning

Logical reasoning is about drawing valid conclusions from given statements — separating what must be true from what merely might be true.

Course progress: 25%

01 Key Concepts

Statements 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.

Syllogisms

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.'

Valid 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.

Blood Relation Puzzles

Problems that require tracing family relationships step by step from a set of given clues to answer a specific question.

Direction Sense Problems

Problems involving movement in compass directions (North, South, East, West), often solved by sketching a simple path diagram.

02 Solved Examples

Example 1 Statements: All roses are flowers. All flowers need water. Conclusion: All roses need water. Is this valid?
  1. From 'All roses are flowers,' every rose belongs to the flower category.
  2. From 'All flowers need water,' every member of the flower category needs water.
  3. Since every rose is a flower, every rose must need water.
Answer: Yes, the conclusion is valid
Example 2 Statements: Some cats are black. Some black things are cars. Conclusion: Some cats are cars. Is this valid?
  1. 'Some cats are black' only tells us there's overlap between cats and black things — not that ALL cats are black.
  2. 'Some black things are cars' only tells us there's overlap between black things and cars.
  3. There's no guarantee the overlapping cats and the overlapping cars are the same black things.
Answer: No, the conclusion does not necessarily follow
Example 3 A man walks 5km North, then 3km East, then 5km South. How far is he from his starting point, and in what direction?
  1. Walking North then South cancels out (5km North, 5km South returns to the same North-South position).
  2. Only the 3km East movement remains.
Answer: He is 3km East of his starting point

03 Practice Questions

1Statements: All dogs are mammals. All mammals are animals. Conclusion: All dogs are animals. Valid?
Yes, valid
2Statements: Some birds can fly. Penguins are birds. Conclusion: Penguins can fly. Valid?
No, invalid — 'some' doesn't mean 'all'
3A person walks 4km East then 4km West. How far from the start are they?
0km, back at the start
4If A is B's father and B is C's father, what is A to C?
A is C's grandfather
5What makes a syllogism's conclusion 'valid'?
It must necessarily follow from the given premises in every possible case

📄 Logical Reasoning — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet