As Regards Formal Logic and Set Theory, what is a Dichotomy?

Introduction:

The mathematical term:

‘dichotomy’

is employed in common parlance, usually as a component of the phrase:

a false dichotomy

.  When we refer to:

a false dichotomy

in common English, we generally refer to two competing ideologies that might seem, at first glance, to have nothing in common; no shared elements; but, upon further inspection, actually have a lot of elements in common.

The Term ‘Dichotomy’ in Mathematics and Logic:

The purpose of this blogpost is to examine what the term:

‘dichotomy’

means in Mathematics and Formal Logic.

The Etymology of the Term, ‘Dichotomy’:

In Ancient Greek, the term:

δίχα

or, when transliterated:

‘dícha’[1].

is an adverb that means:

‘in two,’ ‘apart’

; and the Ancient-Greek term:

τέμνω

or, when transliterated:

‘témnō’

, is a verb that means:

‘I cut,’ ‘I hew’

; and the Ancient-Greek term:

-ια

or, when transliterated:

‘-ia’

is a suffix that is used in Greek so as to form abstract nouns.

When we combine these Ancient-Greek roots together, what we thus derive is the English noun:

‘dichotomy’

, which, etymologically, means:

‘the abstract quality of being cut in two’

or:

‘the state of being cut in two’

.

In Mathematics:

In Mathematics, the term:

‘dichotomy’

refers to a set that can be divided into two subsets that have no elements in common, but - when united in a disjoint union - equal the original set (which is a superset of the subsets).

Should:

A

be our set, then:

A

can be dichotomised into the subsets:

X

and:

Y

where:

X ∪ Y = A

and:

X ∩ Y = ∅

In Mathematics, two criteria must be fulfilled in order that the two subsets formed by splitting the original set, be considered a valid Dicchotomy:

• The two subsets must be jointly exhaustive;
• The two subsets must be mutually exclusive.

The Two Subsets Must Be Jointly Exhaustive:

That is: the elements contained in:

X

along with the elements contained in:

Y

must, together equal all the elements of:

A

, i.e. none must be missing.  Employing the notation of set theory, we write this:

X ∪ Y = A

.

The Two Subsets Must Be Mutually Exclusive:

That is: none of the elements contained in:

X

must be found in:

Y

; none of the elements found in:

That is: none of the elements contained in:

Y

must be found in:

X

.  Employing the notation of set theory, we write this:

X ∩ Y = ∅

An Example: The Set of Colours of a Rainbow.

As an example, let us consider the colours of a rainbow:

    Figure 1:  A rainbow that I scripted in SVG.  The colours of the rainbow are red, orange, yellow, green, blue, indigo, violet.

Let:

R

equal the set of rainbow colours:

R is the set that contains the seven colours of the rainbow.

R = {the seven colours of the rainbow}

R = {red, orange, yellow, green, blue, indigo, violet}

We may diagram the above mathematical statements, thus:

    Figure 2:  R is a subset of the Universal Set.

As we can observe in Figure 2 the set, R is a subset of the Universal Set, or:

R ⊆ 𝕌

.

Let us examine the set:

R

, a little more closely:

    Figure 3:  The set R and its members or elements as a Venn diagram.

In Figure 3, we may observe the set:

R

, as a Venn Diagram, as well as the:

elements

or:

members

of that set.

Let us now dichotomise – or cut in two – the set:

R

, or the set of colours of the rainbow, into the two subsets:

X

and:

Y

, where:

X

is equal to the set of primary colours[2]., or:

{ red , green , blue }

, and where:

Y

is equal to the set of non primary colours, or:

{ orange , yellow , indigo , violet}

.  Hence:

X = { red , green , blue }

and:

Y = { orange , yellow , indigo , violet}

.

Testing to See Whether or Not the Criteria for a Dichotomy Be Fulfilled:

For:

X

and:

Y

to be considered dichotomous subsets of:

R

, then:

X ∪ Y =R

, or:

X Union Y must equal R

, and:

X ∩ Y = ∅

, or:

X Intersection Y must equal the empty set

.

Does X Union Y equal R?

Well:

X = { red , green; , blue }

and:

Y = { orange , yellow , indigo , violet }

and:

R = { red , orange , yellow , green; , blue , indigo , violet }

.  Also:

X ∪ Y = { red , orange , yellow , green; , blue , indigo , violet }

.  According to Euclid's 1^st axiom:

Things which are equal to the same thing are also equal to one another.

Therefore:

X ∪ Y =R

.

Also:

X Intersection Y must equal the empty set.

or:

X ∩ Y = ∅

. Well:

X = { red , green; , blue }

and:

Y = { orange , yellow , indigo , violet }

. We may observe that neither set:

X

nor set:

Y

have any elements or members in common. Therefore:

X Intersection Y equals the empty set.

or:

X ∩ Y = ∅

.

We have thus proven that the set of rainbow colours, or can be cut in two; into two truly dichotomous subsets, which are the set of primary colours and the set of non primary colours.

In Conclusion:

R = { red , orange , yellow , green , blue , indigo , violet }

X = { red , green , blue }

Y = { orange , yellow , indigo , violet }

X ⊂ R

Y ⊂ R

X ∪ Y =R

X ∩ Y = ∅

---

Glossary:

dichotomy

/d^ʌɪˈk^ɒtəmi, d^ɪ-/ noun. (plural dichotomies) [usually in singular.]
1. a division or contrast between two things that are represented as being opposed or entirely different: a rigid dichotomy between science and mysticism.
2. [mass noun] [BOTONY] repeated branching into two equal parts.

‹ORIGIN› late 16th century: via modern Latin from Greek dikhotomia, from dikho- ‘in two, apart’ + ‘-tomia’ (see -TOMY).[3].

---

[1].  From the Ancient-Greek adverb, δίς, or, when transliterated, ‘dís,’ which means ‘twice.’

[2].  In physics, primary colours are those that, when mixed in various combinations, produce all other colours. We are here concerned with light, not with painting, so, for us, the set of primary colours consists of the elements {red, green, blue}.  Televisions and computer monitors use red, green and blue lights so as to produce upon the display all other colours.

[3].  Oxford University Press.  Oxford Dictionary of English (Electronic Edition). Oxford.  2010.  Loc.  107180