1.0 : Introduction to the Basic Language of Mathematics

Introduction

Mathematical objects come into existence by definitions. These definitions must give an absolutely clear picture of the word. We don't need to prove them. We are going to state some basic facts that are needed in this course:

Basic Facts:

The collection of counting numbers otherwise known as the collection of natural numbers is usually denoted by \( \mathbb{N}. \) We write \(\mathbb{N} = \{ 1,2,3,4, \dots\}.\)

The collection of the integers is usually denoted by \(\mathbb{Z}\) and we write \({\mathbb{Z}} = \{ \dots,-3,-2,-1,0,1,2,3,4, \dots\}.\)

The collection of all rational numbers (fractions) is usually denoted by \(\mathbb{Q}\) and we write \({\mathbb{Q}} = \left\{ \frac{a}{b}: a \mbox{ and }b \mbox{ are integers}, \, b \ne 0 \right\}.\)

The collection of all irrational numbers is denoted by \({\mathbb{Q^c}}\).

The collection of all real numbers is denoted by \(\mathbb{R}\). This set contains all of the rational numbers and all of the irrational numbers.

We shall assume the use of the usual addition, subtraction, multiplication, and division as operations and inequalities (\(<,>,\leq,\geq)\)and equality (\(=\)), are relations on \(\mathbb{R}\).

Recall that, if \(a\) and \(b\) are real numbers, then

1. \(a<b\) means that \(a\) is less than \(b.\)
2. \(a>b\) means that \(a\) is greater than \(b.\)
3. \(a \leq b\) means that \(a\) is less than or equal to \(b.\)
4. \(a \geq b\) means that \(a\) is greater than or equal to \(b.\)

Definitions:

1. A real number is called positive if it is greater than \(0\).
2. A real number is called non-negative if it is greater than or equal to \(0\).
3. An integer \(n\) is an even number if there is an integer \(m\) such that \(n=2m\).
4. An integer \(n\) is an odd number if there is an integer \(m\) such that \(n=2m+1\).
5. An integer \(a\) is said to be divisible by an integer \(b\) if there is an integer \(m\) such that \(a=bm\). In this case, we can say that \(b\) divides \(a\) and denoted \(b|a\). Further, \(b\) is called a divisor (factor) of \(a\).
6. A positive integer \(p\) is called prime if \(p>1\) and the only positive divisors of \(p\) are \(1\) and \(p\).
7. A positive integer \(n\) is called composite if there is a positive integer \(m\) such that \(1<m< n\) and \(m|n\).

Mathematical Statements

Thinking Out Loud:

Is \(2 \leq 3\) true or false? How do you know? Can you prove it?

"Knowledge is twofold and consists not only of an affirmation in what is true but in the negation of what is false." -Charles Caleb Colton, Lacon.

In any study of mathematics, language plays a vital role. Mathematical sentences are critical to any mathematical discussion, which are used to express ideas. A mathematical statement is a declarative sentence that is either true or false, but not both. A statement is sometimes called a proposition. The key to constructing a good mathematical statement is that there must be no ambiguity. To be a statement, a sentence must be true or false. It cannot be both. In mathematics, the truth of a statement is established beyond ANY doubt by a well-reasoned (logical) argument. We build upon the truths already established.

So, a sentence such as "The house is beautiful" is not a statement, since whether the sentence is true or not is a matter of opinion.

A question such as "Is it snowing?" is not a statement, because it is a question and is not declaring that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation \( 3x + 5 = 10\) is not a statement, since we do not know what \(x \) represents. If we substitute a specific value for \( x\) (such as \(x = 4\)), then the resulting equation, \( 3x + 5 = 10\) is a statement (which is a false statement).

"There exists a real number x such that \( x^2 + 1 = 0\)" or "\(\exists \, x \in \mathbb{R} \, s.t. x^2 + 1 = 0\)" is a statement because either such a real number exists or such a real number does not exist. In this case, this is a false statement since such a real number does not exist.

Following are some more examples:

Example \(\PageIndex{1}\)

The following are propositions:

• Zero times any real number is zero.
• \(1+1 = 2.\)
• All birds can fly. (This is a false statement, how can you establish that?)

The following are not propositions:

• Come here.
• Who are you?
• I am lying right now. (This is a paradox, if I'm lying I'm telling the truth and if I'm telling the truth I'm lying).

Exercise \(\PageIndex{1}\)

Which of the following are statements (propositions):

1. I like sports cars.
2. \( 2+3=6\).
3. Where are you?

Answer

only the second one.

Notations & Definitions

• A mathematical statement is a declarative sentence that is either true or false, but not both. A statement is sometimes called a proposition.
• \(\exists\): mathematical notation for "there exists".
• \(\in\): mathematical notation indicating a term's inclusion in a set or category.
• \(\mathbb{R}\): represents the set of Real numbers.
• \(s.t.\) and: represents such that.