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Axioms

The term ‘postulate’ was used by Euclid for an assumption in geometry, while the word "axiom" on the other hand, were the assumptions used throughout mathematics and not specifically linked to geometry. Some of Euclid’s axioms, not in his order, are given below :

(1) Things which are equal to the same thing are equal to one another.

      If a = b and a = c then b = c

(2) If equals are added to equals, the wholes are equal.

      If a= b and c = d, then a + c = b + d
(3) If equals are subtracted from equals, the remainders are equal.

      If a= b and c = d, then a - c = b - d
(4) Things which coincide with one another are equal to one another.

      X coincide with X, hence X = X
(5) The whole is greater than the part.

    
(6) Things which are double of the same things are equal to one another.

      If a = b, then 2a = 2b
(7) Things which are halves of the same things are equal to one another.

      If a = b, then

Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.





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