The Proof of the Cancellation Law of Addition

In the preceding posts, I have discussed the Axioms of Real Numbers and the Axioms of Equality. We will use these axioms to prove several basic theorems in Algebra. In this post, we prove the cancellation law of addition. That is, for any numbers a, b, and c,  if a + c = b + c, then a = b.

Theorem

For any real numbers a, b, c,  if a + c = b + c, then a = b.

Proof

a + c = b + c

Adding  –c  to both sides, we have

(a + c) + (-c) = (b + c) + (-c)

Using Axiom 2A, the associativity of addition, we have

a + (c + (-c)) = b + (c + (-c)).

By Axiom 6A, the existence of additive inverse, we get

a + 0 = b + 0.

By Axiom 5A, the existence of additive identity

a = b.

That ends the proof.

What are mathematical proofs, really?

Introduction

In an elementary class, the teacher asked the pupils to do an experiment. He told them to plant flowers and group the flowers into two.  The first group was to be placed outdoors where there is sunlight. The other group was to be kept in a dim room. After a while, the pupils observed that the flowers outdoors grew healthy, while those kept indoors either died or were not as healthy. The pupils concluded that plants need sunlight.

flowers and mathematical proofs

In the experiment, the pupils observed that flowers with enough sunlight are healthier compared to those placed indoor. Some pupils may have an idea (a hypothesis) about this concept based on prior observation. The conclusion that plants need sunlight was based on the observation from the experiment.

In mathematics, similar situations occur. The hypotheses in mathematics are based on observed patterns. For instance, we can “conclude” from a few examples that the sum of two even integers is even. Continue reading