Question

A theorem is:
A.An assumption
B.Always true
C.Always false
D.Sometimes true and sometimes false

Answer 1

Hint: Use the basic definitions of the theorem, lemma and corollary to differentiate all of them and study whether what is the outcome of a theorem.

Complete step-by-step answer:
A theorem is a statement having proof in such a system. Once we have adopted a given proof system that is sound, and the axioms are all necessarily true, then the theorems will also be necessarily true.
So, we can also say that the theorem is always true when the given axioms are satisfied.
Let us consider the example of the Pythagoras theorem, according to the Pythagoras theorem in an right angled triangle, the sum of the squares of the perpendicular and base is equal to the square of the hypotenuse.
In other words, ${H^2} = {P^2} + {B^2}$ where $H$ is the length of the hypotenuse, $P$ is the length of the perpendicular and $B$ be the length of the base of right angled triangle.
So, the axiom in the Pythagoras theorem is that the triangle needs to be right angled and the theorem is always true for these triangles.
So, the correct answer is “Option B”.

Note: The difference between the axioms and theorem is that the axioms are the prerequisite for the theorem and theorem is a result which holds good when the axioms are true.