Practical Work 2 — Relations: Properties, Representations, and Operations #

Second practical class of Computer Discrete Mathematics. Working by hand, you classify relations by the six key properties (reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive), combine relations with set operations and the inverse, and represent a relation three ways — ordered pairs, a Boolean matrix, and a directed graph — to decide its type.

At a glance #

Contents #

Conventions #

• Language — all work is written in English.
• By hand, from the definitions. Classify each relation directly against the definitions of the six properties, and justify every “yes” and every “no” — a “no” needs a specific counterexample (a witness pair).
• Notation is fixed and self-contained. The symbols $A \times B$, $a\,R\,b$, $(a,b)$, $R^{-1}$, and the property names are defined in 2method.md; no outside reference is needed.
• Three representations, one relation. A relation may be written as a set of ordered pairs, a Boolean matrix, or a directed graph; the three are interchangeable, and a task may ask for any of them.
• Every task has a full solution. 3classroom.md gives each task together with a complete worked answer.

Summary #

Relations are how discrete mathematics models “is related to” — divisibility, ordering, equality, adjacency, and the rows of a database table are all relations. This practical drills the core skill: testing a relation for the six properties and reading off its type. The tasks move from classifying six small relations on $\{1,2,3,4\}$, through relations defined by a formula on the integers, to combining relations with union/intersection/difference/complement and the inverse, and finish by drawing one relation as pairs, a matrix, and a graph to recognise it as a partial order. These are the exact skills used for equivalence relations, orders, functions, and graphs in the rest of the course.

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