Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
Axiom 5 states that the whole is greater than the part. This axiom is known as a universal truth because it holds true in any field, and not just in the field of mathematics. Let us take two cases − one in the field of mathematics, and one other than that.
Let t represent a whole quantity and only a, b, c are parts of it.
t = a + b + c
Clearly, t will be greater than all its parts a, b, and c.
Therefore, it is rightly said that the whole is greater than the part.
Let us consider the continent Asia. Then, let us consider a country India which belongs to Asia. India is a part of Asia and it can also be observed that Asia is greater than India. That is why we can say that the whole is greater than the part. This is true for anything in any part of the world and is thus a universal truth.
Give a definition for line segment. Are there other terms that need to be defined first? What are they, and how might you define them?
Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In the following figure, if AB = PQ and PQ = XY, then AB = XY
Give a definition for parallel lines. Are there other terms that need to be defined first? What are they, and how might you define them?
Give a definition for perpendicular lines. Are there other terms that need to be defined first? What are they, and how might you define them?