Sets, Relations, Life, and Deathby Pavel Gladyshev

Mathematically speaking…

Objects

• Chair, You, Me, 1, 2, 3, UCD, pack of pringles

• Any two objects x and y can be compared for equality:

Set

• Unoprdered collecton of distinct objects (members)

• May contain other sets as members

• Set names are written in capital Roman letters: A, B, N, R, …

Membership

Standard sets

Set definition

1. Enumeration of elements of the set

2. Restriction of (i.e. selection of elements from) an existing set

3. Combining existing sets using set operators

Set Definition by Enumeration

Set Definition by Restriction

• A consists of all members of B for which the logical formula is true

Logical formula

• Basic statements

• Connected with logical connectives

If no base set is specified, x is assumed to be a member of the Universal set:

Subset

“A is a subset of B” if every element of A is a member of B.

`

B

`

A

Set equality

Set operators

• Union

• Intersection

• Subtraction

Set operations

• Union:

• Intersection:

• Difference (“subtraction”)

• Assuming that – All men are mortal– Socratis is a man

• Convince me that Socratis is mortal

Tuple

• Ordered sequence of objects• Same object can appear in a tuple several

times• Elements of a tuple are referred to with

subscripts:

a

bc

1

2

3

Set A(Domain of the retation)

Set B(Range of the retation)

Relation

A collection of links between elements of two or more sets:

Relation• Formally defined as a set of tuples:

• Can be viewed as predicate:

Let’s talk mathematically about sexuality

• Define concepts “heterosexual”, “gay”, “lesbian”, etc.

• Convince me using Venn diagrams that gays are human beings.