Understanding Propositions

In the previous post, I have promised that we will take our discussion about mathematical proofs to the next level. We will start this journey by learning about propositions.

In our daily lives, we often encounter statements that are either true or false. Some examples are shown below.

1.) Australia is an island continent.
2.) The Earth revolves around the sun.
3.) Asia is the largest continent.
4.) The sum of 3 and 5 is 9.
5.) The Earth has two moons.

As we can see, the first three statements above are true while the last two statements are false.

Statements that are either true or false, but not both are called propositions. If a proposition is true, then its truth value is True (usually denoted by T) and if it is false, its truth value is False (usually denoted by F). Another characteristic of a proposition is it is a declarative sentence. Therefore, to check if a statement is a proposition or not,  you have to check if  (1) it is a declarative sentence and (2) it is either true or false.  Continue reading

A reboot for Proofs from the Book

I have decided to include undergraduate proofs in this blog. We will be studying formal proofs starting with logic (truth tables, logical connectives, etc.) and then study different methods of proofs (direct, indirect, etc).

While studying these concepts, we will have numerous examples from different branches of mathematics particularly number theory, probability, and combinatorics.

I’m very excited to start this series, so just keep posted.

Understanding If-Then Statements Part 1

In this post, we are going to examine the structure and truth table of the conditional statements or if-then statements. If-then statements are used often in mathematical proofs as well as real-life conversations. But before that, let us understand what proposition means.

A proposition is a declarative sentence that is either True (T) or false (F) but not both. If-then statements are composed of two propositions.

Let us consider the following statement by a father to his son.

“If you get an A in Calculus, then I’ll buy you a laptop.”

We can split this statement into two propositions p and q as follows.  Continue reading