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We're going to start with inequalities, beginning with a short introduction. This lecture

will give you the basic idea behind symbols like < b, x > y and c ≤ d. The first four of

these are really well-defined mathematical concepts.

Inequalities, Basic Ideas

a<b

c<=d

x<y

z=>w

e<<f

Let's start from the top and draw a real number line. And suppose here is 0, here is

the number 2. here is the number 3.1, and let us consider this statement: 2 < 3.1.

This funny little pacman symbol here pointing that way means "is less than." So here

we have a statement; let us think about what it means and see why it is true. This

statement: 2 < 3.1 means "2 is to the left of 3.

1 on the real number line." Let's say

11.78 < 3.1; say it's about there. Fact: 11.78 is also to the left of 3.1 on the number

line (on a real number line). In general when we write A less than B, we mean A

wherever it is on the real number line (on a real number line).

The symbol < denotes less than. There is no universally accepted symbol for greater

than. It can be written as >, or 2, or sometimes it is not written at all. When two

numbers are compared using S, it means that the first number (on the left) is less

than the second number (on the right). For example, 3.

1 <2 means that 3.1 is less

than 2. The other comparison operators are: >, >=, and ≤ .

[Graph]

Here's this funny symbol, x << y. And you actually never see this in a proper math

textbook, because it's not really a proper mathematical concept. But you'll see this all

the time in data science. What this really means x is much, much less than y. So for

instance, the statement 1 < 1,000,000 might be a reasonable thing. I would argue

that it is not true. It is not possible for a judge to determine whether this statement is

true or false. Whereas a statement like 2 < 3.1 can be determined to be either true or

false. It is much less than. It's in the eye of the beholder but we tend to agree on

what it means. Thus far we have considered statements with greater than and less

than statements: now let's think about equal to statements.

[Graph]

To write the mathematical symbol "<=" as "less than or equal to," one should first

write the less-than sign and then place a horizontal line below it. One also

sometimes sees this symbol written as "a < b". This means that a <b, or a = b. It's

just shorthand way of saying that either a is less than b. or a is equal to b. If one

wants to write the sentence "a is less than or equal to b", it can be written two ways:

"a <b" or "a ≤b".

When a claim is made, you can test it by trying to disprove it by showing that one of

its statements is false, while the other is true. For example, suppose someone claims

that 2 < 3.1. This means that either 2 < 3.1, or 2 = 3.1. The former statement is false,

since 2 > 3 Therefore, the person's claim 1S true only if he meant the latter

statement and said that 2 = 3.

[Graph]

Suppose someone claims that 2 is less than or equal to 0.8. This means either 2 <

0.8 or 2 = 0.8. However, both statements are false and so the claim is itself false

overall.

This concludes the lecture on basic inequalities. We've learned what less than

means, greater than, less than or equal to, and greater than or equal to, all of which

are standard mathematical concepts. We've also learned what it means to say that

A <B, which is not a standard mathematical concept but one that people use

informally.

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