Transitive Relation

Learn

If A > B and B > C, then A > C. Applies to ordering relations.

A transitive relation is one where if the relation holds between A and B, and between B and C, then it must hold between A and C. Common transitive relations include 'greater than', 'less than', 'older than', 'taller than', 'ancestor of', and 'implies'. Recognizing transitivity allows us to chain comparisons and orderings together.

R(a,b) ∧ R(b,c) → R(a,c)

• Alice is taller than Bob. Bob is taller than Carol. Therefore, Alice is taller than Carol.
• 5 > 3 and 3 > 1, therefore 5 > 1.
• New York is larger than Chicago. Chicago is larger than Houston. Therefore, New York is larger than Houston.
• Iron is heavier than aluminum. Aluminum is heavier than plastic. Therefore, iron is heavier than plastic.

How to recognize it

• Look for comparative statements using the same relation (taller than, faster than, older than, etc.)
• Check if there's a 'middle' element that appears in both comparisons
• The conclusion compares the 'outer' elements using the same relation

Common mistakes

• Not all relations are transitive. 'Is a friend of' is NOT transitive (A's friend's friend may not be A's friend)
• 'Is next to' is NOT transitive (if A is next to B, and B is next to C, A may not be next to C)
• Applying transitivity to non-transitive relations leads to invalid conclusions

Practice guidance

Logical Deduction