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1. “Things which are equal to the same thing are also equal to one another.”
Explanation:)
Euclid’s first axiom asserts that equality is transitive. In algebraic logic, one would say “if x=y, and if y=z, then x=z.”
2. "If equals are added to equals, the wholes are equal."
Explanation:)
One reason why Euclid had to separate his axioms regarding addition and subtraction, is that Greek mathematics did not include negative numbers.
An algebraic version of Euclid’s second axiom would read “if x=y, and if a=b, then x + a = y + b.”
3.
(6) Things which are double of the same things are equal to one another.
Example : If 2x = 2y then x = y.
Explanation:)
Euclid’s first axiom asserts that equality is transitive. In algebraic logic, one would say “if x=y, and if y=z, then x=z.”
2. "If equals are added to equals, the wholes are equal."
Explanation:)
One reason why Euclid had to separate his axioms regarding addition and subtraction, is that Greek mathematics did not include negative numbers.
An algebraic version of Euclid’s second axiom would read “if x=y, and if a=b, then x + a = y + b.”
3.
(6) Things which are double of the same things are equal to one another.
Example : If 2x = 2y then x = y.
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