Set Theory 1 | | DJS

Set Theory 1

The study of set theory has been lost from various syllabi. This should rectify the situation and I will (did) teach this (simple) stuff to year Eight from 2006. There are lots of new symbols to digest, some of which are hard to reproduce,  so this page may be slower to load than the others.

A set is a collection of things, called elements or members of the set. It is convenient that we use the same words for classes; you are a member of one of my maths sets. If A is a set and x is an element of that set, then we say x belongs to A or “x is an element of A” and this is written x∈N. If x is not in A then we write x∉A, or put an oblique (/) line through the  symbol. Those were  Unicode U+2208 and 2209.

Two sets are equal if they contain the same elements – every element of A is in B and every element of B is in A. If every element of a is in B but B has more elements than A, we say A is contained in B, written A⊂B, and if A⊆B it is possible that A = B too. It is possible that a set has no elements, which would make it an (the) empty set, written ∅ or {}. A  B defines A as a subset of B, which includes ⊆ as a possibility. B⊃A (or B⊇A) also says A is a subset of B but should be read as “B is contained in A”. The elements of a set generally have no order and are written {inside brackets like this}.


1  Let N be the set of natural numbers {1,2,3,4…….}. Then 4∈N, SusieN, -8N, 0.5N, 0N are all true.

2  Let P be the set of members of the school, including staff. Then DJS∈P, you∈P, Headmaster P and 0∉P, Prime Minister∉P, Julius Caesar∉P are all true. Or they were when I first wrote this.

3  Let A be the set {a,b,c}. Then a∈A, {a,b}∈A,  4A,  d∈A are all true.

We can count the elements of a set such as A above, written |A| or n(A). Think of |A| as “the size of A” and n(A) as “the number of elements in A”. This count is properly called the order, and the order of A in the last example is |A| = 3.
Sets can be
finite (we can count them and state the count) or infinite. The set of natural numbers N is infinite. The set of all integers,  ℤ, is infinite; contains all the positive and negative whole numbers, including zero. I can write = {x | x is an integer}, and you read this as “ℤ is the set of things called x such that x is an integer”, much the  same as “ℤ is the set of all integers”. In this case the symbol to the left of the vertical line, |, is the character to be used for a typical member of the set and the other side of the line is a description of this typical element. For example, R = {r | r is a reader of this page} should include you in R and most of the members of R are also in P as in Example 2.

Other classes of number like ℤ exist, such as ℕ, the natural numbers; ℚ, the rationals; ℝ, the reals;  ℂ, complex numbers. These are all written with an extra penstroke similar to the character I have chosen.

The implication sign =>  or ⇒ is very useful, read as “implies”. For example, if B⊂A (B is contained in A) then x∈B ⇒ x∈A is pretty obvious. “is implied by” is ⇐ and both , “implies and is implied by” is ⟺, better understood as “if and only if” written by me as ‘iff’. So the statement that sets A and B are equal is matched by a mathematical statement such as x∈A ⟺  x∈B.

Last, we frequently use the terms union and intersection. Definitions go like this:

If A and B are sets then the intersection, A∩B, is the set of all elements which belong to both A and to B, i.e. x e A∩B  means x∈A and x∈B. The union is x∈ A∪B so  x ∈ A or x ∈ B. Be careful with your understanding of ‘or’ here; | AB | = | A | + | B | - | AB |  (because AB has been counted twice). So  x∈ A∪B =>  x ∈ A only x ∈ B only or possibly lies in both; it is in the union of the sets. This is the ‘inclusive OR’, and I suspect that people who write management questionnaires do not recognise this, though since most of such people are apparently Americans, maybe it is a convention that starts there. I mean the sort of questionnaire that is allegedly measuring you ability to ‘fit’ with a company.

The union [] is all of the elements in A and all of the elements in B with none counted twice. The intersection [∩] is only those elements simultaneously in both sets.

Please be aware that very many people get these two confused; a union includes everyone and an intersection is the American word for the tiny piece of tarmac on both roads. Unions are big and intersections are small, even when they're big intersections.

Example 4:  Let A ={a,b,c},  B = {d, e, f, c }. Then  A∪B = {a,b,c,d,e,f} and  AB = {c}

You do these:

Let C = {b,c,d,f,g,h} and use A & B as before. List the elements of  A∪C, A∩C, C∪B, B∩C and try to do (A∪B)∩C, A∪(B∩C), (B∪C)∩A. Many people find it helpful to draw a Venn diagram – overlapping circles representing A, B and C, with the elements scattered in the appropriate spaces. I would have shown examples of this in your lesson.

Two more: 

(i) The larger set from which elements may be chosen is usually indicated with an ⋿ or ξ, sometimes, for easier typing, just E. Think of this as E for elements, as in “this E gives you the set of elements I am about to use”.
Example: Let ξ be the counting numbers below ten, so ξ = {1,2,3,4,5,6,7,8,9}

(ii) we have the wonderful ‘not’, a simple dash, so notA is A’, notB is B’. Some textbooks will use the unfamiliar symbol ¬ for 'not', so not A is written ¬A. for a load of things 'not', ise brackets; ¬(A∪B)
Example: Let ξ be as the last example, let A be the even numbers in ξ, then the set A’ is {1,3,5,7,9}.

DJS 20130102

Improved presentation with unicode symbols in update done 201302 and formatting changes in 201304. Small errors fixed  201306. retyping of bits and renewal of strange symbols made  in transfers to new files in 2017. This process repeated in 2021.

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