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Set Theory

Counting Basics (variables and set theory)

Counting basics introduces variables using set theory. From there we move on to property laws and better examples of algebra. Math after arithmetic is called “higher math”. In higher math we categorize numbers by type into groups called sets. The study of this is called set theory. “Counting basics” uses a simple introduction to set theory.

Before 0 or negative numbers were recognized by western, math numbers began at 1 and counted upwards. We call this system the “natural numbers”. Natural numbers are counting numbers of the set (1,2,3...). When you see numbers in a set they are written in either set notation or as a series. A series is a pattern of numbers separated by commas inside parentheses to group them together as a set. Set notation uses brackets [] and () to show what numbers are in a set. The numbers that are in the set are called the domain. Numbers that are not in the set are not in the domain.

The set of counting numbers are whole positive numbers that start at 1.
Remember that 0 is neither positive nor negative so it is not included.

The set of whole numbers is: (1,2,3,4,5,6,7,8,9…) The three dots are called “an ellipsis mark” or “ellipsis”. It means that the numbers go on forever. If we continue the series the numbers go forever by adding +1 to then number before the ellipsis.

If we start at 0 we have a different set called the “whole numbers”. Whole numbers are positive whole numbers. The set of natural numbers is: (0,1,2,3,4,5,6,7,8,9…)

If we start and 0 and either add 1 forever or subtract one forever we have all the positive and negative whole numbers. This set is called “integers”. Integers are whole numbers that can be positive or negative. This is the set we use most often in math. Because the numbers go in both direction forever we put ellipsis in the beginning to show that the series continues backwards forever. We are interested only in set integers and reals. The set of integers uses two ellipsis: (…–4,–3,–2,–1,0,1,2,3,4…)

The next set is fractions also called “quotients” because a fraction bar is a division bar. Fractions are made by taking a whole number and dividing it by a whole number to get a fractional number that is not whole. Fractions can be negative like –¼, mixed like 22½ or improper like 5/4. A fraction can be written like a fraction or a decimal or even a percent. The mixed fraction 22½ can be decimal like 22.5 or improper like 45/2 .

There are proper fractions, improper fractions, mixed fractions, and decimal fractions. All can be negative and all can be converted back and forth from a fraction to a decimal or percentage.

A proper fraction: two whole numbers divided using a “/” that is less than “±1”.
An improper fraction: a mixed fraction written as a fraction instead of a decimal.
A mixed fraction: an improper fraction written as a decimal.
¼ and –¼ are proper fractions because they are part of a number.
45/2 and 5/4 are improper because they are mixed.
We could say “mixed improper fractions” but that is redundant.
The set of fractions is: (all whole or negative fractions). There are fancy ways to write this that come up later.

The next set is “irrationals”. These are square roots and other radicals. Names like irrationals, radicals, roots, or even surds imply a rooting operation; we call these radicals for the radical symbol √. A surd can be considered an irrational square root that has been operated on by another number such as √2 – 1 = 0.414213... . This isolates the irrational part from the integer. We call decimals either decimals or the mantissa.

There are the “real numbers” called the “reals” and they are all numbers except a unusual one called “complex numbers”. Finally the last set is call complex numbers which uses the letter i = √–1. Complex numbers are not used until last part of Algebra for graphing and conics or hyperbolas. There are important ways to write the sets but that requires a study of set theory. Sets can be written dozens of way. Some more useful than others depending on what you want to convey. Instead of defining sets using formal logic I want to present a more informal definition using blackboard letters.

Blackboard letters are also called “double struck” letters because they have double lines. On blackboards they were written this way so that people did not mistake them for variables or other letters. The blackboard letters are the capital funny looking letters. Every set is a superset of the subset before it:

The colored letters match the colored word in either meaning and/or show why that letter is chosen. The blackboard letters are the first letters of the word they represent. These 7 groups are basic and all the groups that exist unless somebody makes up there own group and gives it a letter. Algebra uses the integers and reals. You are not expected to know this or memorize these sets so that Is why I am telling you. Focus on the red sets.

Any set can be represented with a number line. Tools to find numbers, explain concepts or derive answers will use either a number line, a Cartesian graph, or a unit circle. With start with a number line to perform arithmetic functions. Then move on to graphing with uses two number lines, a horizontal number line called the x-axis, and a vertical number line called the y-axis. After Algebra concepts are mastered we introduce geometry with the use of the unit circle. The unit circle is a circle drawn on a Cartesian graph. From geometry we introduce trigonometry and graph triangles in the unit circle. Following basic trigonometry comes advanced definitions of trig functions shown on a Cartesian graph. These definitions help lead to pre-calculus and calculus topics.

The algebra concepts we want to focus on will be variables, functions and formulas. In geometry the focus moves from to construction problems involving area and volume formulas. In Algebra1 we use integers, fractions, and radicals. This set of all these sets is called the reals. In Algebra2 begin to cover complex numbers, conics sections, and sequences.

A set is a group of elements. A set can contain other sets. A set in a set is called a subset, meaning under the set above it, meaning one level beneath it. A superset is the set above a subset. A superset contains smaller sets. A subset in inside a superset. If we want to express infinity on a number line we use the symbol ∞. It can be –∞ for negative infinity or ∞ for positive infinity. If we wanted to graph all the numbers in existence:

If we add move to the right. 2+3 =5 so we start at “2” and move three units to the right stopping at “5”. If we subtract we move to the left. 5 –3 = 2 so we start at “5” and move three units to the left stopping at “2”. People confuse the sign of the number with the sign of the operation. In both examples all numbers are positive quality so they have positive value. This places emphasis on position rather than simply figuring out the final total.

The are special rules for dealing with negative numbers called “value”, “absolute value”, and “sign negation”. There is a special rule for subtracting negatives called “canceling”. Canceling or cancellation or negation mean similar things and there are different situations when these words are applied. We cancel an operation or cancel a value among other things. For subtraction remember this rule: “Two negatives cancel”.

If 4–1 then 3, if 4 – (–1) then 5. When we have two negative signs they cancel and turn into a single plus sign.

Sign conflicts only care about 1 thing: “Do I add or subtract?” which is “Do I move left or move right”. Whatever is before the conflict is irrelevant, only the two signs, operator sign and subtrahend sign, matter. A “minuend” is the thing to be subtracted from which the subtrahend is the thing that does the “subtracting”. They are also both called sums: sum – sum = minuend – subtrahend.

Do not worry about the chart or attempt to memorize. The focus is the blue options where we cancel negatives. Later during polynomials all terms whether (–) or (+) are put in () and added using a + sign. Learn to understand why signs cancel, the difference between quality and quantity, and learn not to confuse “the sign of the operation” with the “sign of the number”.

A positive number like 1 is written without a sign and we still know it is “positive 1”. A negative number like –1 is written with a minus sign to tell us it is “negative 1”. This a “signed number”. You have either a positive or a negative which some people call a number or a signed number. 1+1 both numbers are positive and the operation sign is positive. We isolate the numbers from the operation sign like this: (1)+(1) see how the operator is excluded?

4 – 1 Both numbers are positive but some people and even teachers describe this as only 4 is positive.
(4)–(1) this proves both are positive because the sign is excluded.

In 4 –1 people think 1 is –1 but it isn't. This confusion leads to getting a bad intuition.
There is a difference between four minus one and four minus negative one.

“four minus one” = “4 – 1”
“four minus negative one”= “4 – –1”

If you ever have trouble with any equation try writing it as an English sentence. This is an important life skill[1]. Part of math is getting the answer the other part is communicating it. Four parts:

Learning to write the correct equation in math symbols.
Learning to solve the equation.
Learning to articulate the problem in English writing.
Learning to articulate the answer in English writing.

The chart contained 8 different situations but highlighted in red was when two negatives canceled. Combinations are found using “combinatorics”. Combinatorics would know that there are 8 choices because we have two different places, two different operators, and two different qualities. This gives us 2∙2∙2 choices or 23 which is 8. It does not matter if you start with a negative or positive number and whether the final answer is negative or positive. The only consideration is if you are adding or subtracting from that number. This limits our choices:

The rules above do not change if the sum before is negative. A sum implies a single number or subtotal. Only negative numbers have value signs in front so that is why they are called signed. If there is doubt group numbers with () leaving only one operator between two numbers.

These last two pages can be summed up in if signs same add and if they differ subtract but time was spent to demonstrate why and to prove whether a number is positive or negative not just what direction to move on the number line. This ends proof. â– 

The black dot is used at the end of proofs so say “end of proof”. You do not need to use both. This proof is important because logic and Boolean operators do weird adding of 1 and – 1.

We would never encounter a situation where 1 –+1 existed because we do not write positives with a plus sign. If we group (1)–(+1) the + sign cancels by itself leaving (1)–(1) which is 1–1.

[1] The part about writing your problem as an English sentence applies to non math as well. Before you can solve a life problem you need to define it first. Writing makes people smarter and focus more. It stimulates brain growth. Identify your problem, isolate it, then solve it. This works for all math and life problems. Write out your problems identifying what they really are, what causes them, and what steps you can do to solve them. Make a list of possible solutions then be honest and list the pros and cons for each solution. If you do not do this you will not ever change your predicament. If you are not honest it defeats the purpose. The truth does not care about your feelings and telling yourself what you want to hear is lazy and foolish. If you lie you are in denial. Feelings and emotion create bias that clouds judgment. Notice how your feelings prevent you from accepting what is right for you. Be fair and accept how you really feel but do not let that prevent you from making clear and sound decisions. Recognize bias and make sure you can function and treat people reasonably or else you are not really doing math but pseudo math. We use this in ethics and statistics to apply a fair culture of learning and working with the benefit of all members of society.