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  • Read the following axioms: (i) Things which are equal to the same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) Things which are double of the same thing are equal to one another. Check whether the giv

Read the following axioms:

  1. Things which are equal to the same thing are equal to one another.
  2. If equals are added to equals, the wholes are equal.
  3. Things which are double of the same thing are equal to one another.

Check whether the given system of axioms is consistent or inconsistent.

Answers (1)

Given the system of axioms is consistent.

Solution:

Euclid’s axioms:

  1. Things that are equal to the same thing are also equal to one another
  2. If equals are added to equals, then the wholes are equal
  3. If equals are subtracted from equals, then the differences are equal
  4. Things that coincide with one another are equal to one another
  5. The whole is greater than the part.

Hence we can see (i) is an axiom. (same as axiom 1)

Hence we can see (ii) is also an axiom. (same as axiom 2)

(iii) Yes, it is true that the things which are double of the same thing are equal because it is the conclusion of one of Euclid’s axioms.

Euclid’s first axiom states that things which are equal to the same thing are also equal to one another.

Hence, the things which are double of the same thing are equal to one another.

Hence the given statement is true.

Therefore, the given system of axioms is consistent.

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