MATHS: Introduction + Set Theory

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INTRODUCTION:
In computational design or parametric design. Maths are necessary in order to develop models, resolve problems, optimize and design. The Maths are essential if you want to play in this league. Part of the credit in good designs goes to the artistic and conceptual idea. But the correct use of the maths is the responsible for turn the idea in a project. In a certain sense we use the maths to explain the idea, we need them to improve the idea, we need them to build the idea and to tell others how much will cost our idea. Since the architect profession is being affected by code, software, Bim, Robotics and in general technology: The maths and code are the current alphabet.

SET THEORY:
My school teacher explain me that a Set is an unordered group of unique things. She told us that a set don`t have any particular order for example ‘5’ can be before ‘2’ and ’10’ before the ‘1’. She said also that a set can’t contain two or more copies of the same thing. And a set can be composed about anything: letters, names, persons, cars, trucks, numbers, images, etc. A set is like a box where you can place anything that you want, don`t forget that the items are unique and unordered.
Sets are denoted by curly braces “{}” and we use the commas “,” to separate the items in the set. For example:
The Set of all integers from 1 to 10 inclusive should looks like this: {2, 1, 4, 3, 5, 7, 6, 9, 8, 10}.

Also you can give sets name so we can talk about them without having to list all their items. Usually can name a set by specifying the name, followed by the equal sign (=), like this: my favourite Integers = { 6, 9, 7, 10, 3, 5, 100, 1000}. But in general sets are always named by a single Upper-case italic letter such as: A or B.

Some sets are too large to describe explicitly. for example the set of all integers. In this case our set is an infinite list so we could denote like this: A = (…, -3, -2, -1, 0, 1, 2, 3, …). the previous denotation works fine for some cases, but not all. For example for all Chileans, or the set of real numbers is the same case. In that case instead of listing the items we list the properties of the set. When there is more than one property we can use a comma to separate them or you can use the word “and“, like this P = {the set of all Chileans abroad, over 20 years}.

A variation of this notation is the symbol “|” and one or more variables (symbols that do not have a fixed value). This symbol (|) is read “such that”. Variables are like place-holders for the things being described. For example the set of all beams with a span smaller than 20 Mt.
Q = {x | x is a beam and the radius of the beam is less than 20 Mt.}
You can read this as “Q is the set of all x such that x is a beam and the span of x is less than 20 Mt.” Since x is just a temporary name (Variable), is doesn’t matter how you call it, you could have another symbol “y” or “b” and the set will be the same. Same as coding ;).
Some sets are important enough to have a special name. for example the null set is represented by the symbol ∅, contains nothing, hence its name. Other important set is the universal set that contains everything or everything that can be named, usually is denoted by the symbols ∪ or Ω “omega”.

Other sets with special names are:

R THE SET OF ALL REAL NUMBERS
Z THE SET OF ALL INTEGERS
N THE SET OF ALL NATURAL NUMBERS (0, 1, 2, 3, …)
Z+ THE SET OF POSITIVE INTEGERS (1, 2, 3, …)
Z- THE SET OF NEGATIVE INTEGERS (…, -3, -2, -1)

SET MEMBERS:
When something is in a set, we call it an item of that set. We denote that thing is a member of a set using the symbol ∈, which you can read like ” this is an element of this set”, and then the set name. A is an element of the Alphabet set. Also you can denote that a thing is a not member of a set in a similar way, using the symbol ∉. This element is no an element of the set… For example.

x ∈  { x, y, z}

12.456 ∉ {x | x is an integers}

SUBSETS, SUPERSETS, EQUALITY
Sometimes all items in one set are contained in another set. For example, the set of integers is contained in the set of all real numbers. we can said the one set is a subset of the others. If set A contains the set B we can expressed like this: B⊆A which means, A is a subset of B. Any given set contains all of its own members, therefore every set is a subset of itself. the subset operator is very similar to the less than or equals operator (<=), because every number is less than or equal to itself just like every set is a subset of itself. For example.

1 <= 5  <= 9 ∴ 1 <= 9 //If A is a subset of B and B is a subset of C then A is a subset of C.

John Venn invented the Venn diagrams to visualize the relationship between sets. Usually represented by circles, the geometric points inside the shape are the members of the set. When two sets A and B  contain exactly the same items, we can say the sets are equal, and denoted it A = B. is possible to very that break down the problem. first prove that the first set is a subset of the second set and then prove the second set is a subset of the first set.

ALGEBRA for SET THEORY

We can say a operation is commutative if we can rearrange the order of the terms without changing the of the expression. Multiplication and Addition of real numbers are both commutative: a + b = b + a, and ab = ba. An operation is said to be associative if you can add parentheses any where without changing the value of the expression: (A + B) – C = A + (B – C). Finally an operation “X” is distributive with respect to an operation “+”: a(b + c) = ab + ac. the difference is that they are not applicable  for numbers or items in a set. but the operators like addition, subtraction and multiplication. keep in mind the multiplication is distributive with respect to addition.

The main operators in set theory are: Complement, Union, Intersection, Difference, Cardinality and Cartesian product.

The complement operator denote everything that is not in a given set; is similar to the negation operator (!, in C#). Is the set that contains everything in the universal set that is not in a given set A is called the complement of A. Usually denoted Aor A’ or Ā: A complement. One of the properties of the complement operator is that the complement of a complement of a set is that set itself. (Ac)= A. Same in algebra: -(-2)  is actually +2.

The union operator is a way of combining sets, and as such, is similar to the addition operator. The union of two sets is a set that contains every item in both sets. A∪B, A union B. A∪B = B∪A, so we can say the union operator is commutative.

The intersection operator is a set that contains every item that is in both A and B, and nothing else. If the item is in one set but not the other, the it is not in the intersection of the two sets. A∩B A intersect B

Difference operator is similar to the subtraction operator for real numbers. the difference of A and B, A-B are all the items contained in A but not in B. The difference operator is associative but not commutative A-B is not, usually the same as  B-A. The only time this will be true is when both sets are equal, in which case A = B and the difference is the null set.

The cardinality operator is the number of items in that set. The cardinality only is applied to finite sets. no reason to try count infinite elements. Cardinality is denoted like set A as |A|.

 The result of cartesian product operator is a set of ordered pairs. An ordered pair is in the form (x, y). The pair are referred to as “ordered” bacause the pair (x, y) is different as the pair (y, x). The cartesian product of A and B, AxB, is a set of ordered pairs such that the first element in the ordered pair comes from A and the second element comes from B.

AxB = {(x,y)| x ∈ A, y ∈ B}

set theory scheme-01

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