Sets, Relations

Very many computer data structures are best reasoned about using things called ``sets''. For the purpose of this course a set is collection of things, and is written by listing the members of the set inside curly brackets.. So for instance the set whose members are the first five whole numbers might be written {0,1,2,3,4}. It is perfectly possible to have a set with no members at all, and it is then (obviously) written as {}, and known as the empty set. There are a number of rules about sets and a collection of things you are allowed to do with them:

• The members of a set should not be thought of as being there in an particular order, even though when you write down a representation of a set you have to list them somehow. So for instance {1,2,3}, {3,1,2} and {2,1,3} are all just different ways of writing down the same set.
• Any particular object can either be in a set or not. The consequence of this is that when you list the members of a set you should never see any duplicates. It is not possible for an object to be a member of a set twice or more times, so {1,1,1,2,2,3} is not a valid things to write.
• As well as having the null set {} it is (of course) quite proper to have sets that have a single member. Set members can be arbitrary things, including other sets. Thus we can have the number 1, and the sets {1}, {{1}}, {{{1}}}. These are all quite different things. For instance the set {{}, {{}}} is a set with two members (one of which is the empty set, the other is a set whose sole member is the empty set).
• It is quite common to use upper case letters to stand for sets and lower case ones for the items that may be members of those sets. The notation $x \in A$ is used to indicate set membership. x is an object and A is a set. Eg $3 \in \{1,2,3,4,5\}$ is true. The notation $A \subset B$ is used to indicate that A is a subset of B, ie it has as its members some selection from the members of B. Eg $\{2,4\} \subset \{1,2,3,4,5\}$.
• The notation $A \cap B$ is a set whose members are just those things that are present in both A and B (the intersection), while $A \cup B$ has all the members that are in either (or both) A and B(the union).
• The exact rules for the treatment of very infinite sets are not needed in this course, but informal notation will be used to describe simple cases, such as the integers $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$. The expression $\{x^{2} \mid x \in \{0, 1, \ldots\}\}$ stands for $\{0,1,4,9,\ldots\}$.
• Given a finite set you can find out how many members it has. The number is known as the cardinality of the set (or more simply as its size), and is sometime written |A| as in $\vert\{2,4,6\}\vert = 3$.

An ordered pair is just a pair of things grouped together where (unlike the situation with sets) the ordering of the two items does match. Such pairs will be written with parentheses as in (1,2) rather than curly brackets. This idea will be extended to ordered triples, quadruples, 5-tuples and so on, and in general n-tuples. The objects in an n-tuple do not need to be all of the same sort. Eg here is a 4-tuple

    (1, "string", {{}, {22}}, x}

where the third member of the 4-tuple is a set.

Given one or more sets there are ways of constructing bigger sets out of them. The cartesian product of two sets A and B is the set of ordered pairs (a,b) with $a \in A$ and $b \in B$. Eg $\{a,b\} \times \{1,2\} = \{(a,1), (a,2), (b,1), (b,2)\}$. The powerset of a set A is the collection of all subsets of A. Eg. powerset($\{p,q,r\}$) $= \{\{\}, \{p\}, \{q\}, \{r\}, \{p,q\}, \{q,r\}, \{p,r\}, \{p,q,r\}\}$

You might like to convince yourself that $\vert A \times B\vert = \vert A\vert \vert B\vert$ and |powerset (A)| = 2^|A|. For the powerset example try induction on the size of the set.

Given a set X , a relation is some property that may or may not hold between one member of X and another. For instance if attention is restricted to sets of numbers then the operator ``<'' for ``is less than'' is a valid relation, as would be ``='' for equality. Sometimes people will want to use a general name, say R for an unspecified relation, and then rather than something concrete like x=y they will write xRy to show that x relates to y under R.

Relations are not only things that arise with numbers. In a family tree (or many computer data structures you will come across later) the relation ``is an ancestor of'' can be relevant. Given a set of people one could specify the relationship ``likes'' (useful for a computer-based system for arranging seating plans at large dinners?). A final example is the ``is married to'' relation.

Relations can have different properties, and the examples given above can illustrate some of the important ones:

Reflexive:

Given any member of the set, x say, does the relation hold true between x and itself. For = it does, for < and ``is married to'' it does not and for ``likes'' the status is unclear to me. Relations that have this property are known as reflexive. From any (possibly non-reflexive) relation you can derive something called the reflexive closure by forcing each item x to relate to itself but otherwise leaving conditions unchanged.

Symmetric:

Of the examples given here, = and ``is married to'' have the property that if (x,y) relate then (y,x) do too. This makes them symmetric, while < is clearly not. Again the extent to which ``likes'' is a symmetric relation on any particular set is an interesting social consideration. The reflexive closure of a relation extends a relation to force symmetry. Eg the symmetric closure of < is $\neq$.

Transitive:

If x=y and y=z then we may deduce that x=z. Similarly for <, and these relations are, on account of this, known as transitive. There is such a thing as a transitive closure, which is discussed in the next section.