Chapter 15 talks about the topic of proofs in mathematics. It is often said that proof is the foundation of mathematics and therefore in this chapter students will look at a few aspects of proof as well as develop an idea of what exactly are mathematical proofs. The chapter also tries to explain the meaning of a mathematically acceptable statement. Some of the other topics discussed in this chapter include;

- What a statement is.
- What kind of reasoning is involved in mathematics?
- What a mathematical proof consists of.

## Mathematical Proofs

A mathematical proof is a process which can establish the truth of a mathematical statement based purely on logical arguments. Each proof is made up of a successive sequence of mathematical statements.

Statements are assertive sentences that can be judged either true or false on some criteria by some process. Mathematical statements are of a distinct nature from general statements.

### Some Related Terms

Axioms are statements without any proof and are assumed to be true.

A conjecture is a statement which is believed to be true based on some mathematical intuition, but which we are yet to prove.

A theorem is a mathematical statement whose truth has been established or proved.

Students can check out some of the chapter questions along with their solutions to know more about the concept.

**Question 1:** State whether the statement is true; “Product of two odd integers is even”.

So we will take two let’s say 5 and 7 as the odd integers. Normally their product is 35, which is an odd integer again. Thus, we can say that the statement is false.

However, the main point here is that the statement is justified with an example that runs counter to the statement.

**Question 2:** Prove that the sum of any two even integers a and b is even.

**Solution: **Since it is given that a and b are even 2 can be taken as a factor. As such we can write;

Let a = 2x and b = 2y for some integers

Then, a+b = 2x+2y

= 2(x+y)

Taking the distributive property of integers, the number 2(x+y) has 2 as a factor. It means that it is even.

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