TeXipedia

therefore

Indicates a logical conclusion or consequence in mathematical proofs and logical arguments.

Overview

Essential in formal mathematical writing and logical deductions where a conclusion follows from previous statements or premises.

  • Commonly used in geometry proofs to show that a conclusion follows from given facts
  • Appears frequently in formal logic and mathematical reasoning
  • Helps structure mathematical arguments by clearly marking the transition from premises to conclusion
  • Often paired with 'because' or 'since' statements in mathematical discourse

Examples

Logical conclusion in a mathematical proof showing that a triangle is isosceles.

∠A=∠B and AB=AC∴△ABC is isosceles\angle A = \angle B \text{ and } AB = AC \therefore \triangle ABC \text{ is isosceles} 
\angle A = \angle B \text{ and } AB = AC \therefore \triangle ABC \text{ is isosceles}

Deductive reasoning in set theory showing a set is empty.

A∩B=∅ and A⊆B∴A=∅A \cap B = \emptyset \text{ and } A \subseteq B \therefore A = \emptyset 
A \cap B = \emptyset \text{ and } A \subseteq B \therefore A = \emptyset

Conclusion of a basic algebraic proof.

x2=4 and x>0∴x=2x^2 = 4 \text{ and } x > 0 \therefore x = 2 
x^2 = 4 \text{ and } x > 0 \therefore x = 2