Compare and contrast single-member districts and proportional representation (PR). How effective is each electoral system in representing its people’s interests? How do they support the party systems around which they arose?

With single-member legislative districts, often employed in "first past the post" voting systems as well as multi-round systems, the whole chamber of a parliamentary body is divided into a number of constituencies. Each of these is represented by a single delegate who represents the majority of the constituency.
Proportional representation usually allocates delegates to a legislative body as a percentage of the total number of votes received by a party. Several different methods for conducting elections under a proportional system exist, including the D'Hondt method, the Single Transferable Vote (STV), and others. Allocation of seats can occur either through local lists (where constituencies are represented by multiple delegates and allocated among parties in proportion to the number of votes received by the candidates from each party) or national lists (where there are no constituencies and a national slate of candidates is selected).
The single-member district system gives a strong voice to the ideological majority within any single district but also promotes greater cohesion and unity by forcing candidates to appeal to the widest swath of voters possible, aggregating as many issues and concerns as they can in the quest to maximize their vote totals. The single-member district may sometimes encourage greater party independence of politicians if a system de-couples a candidate from a party machine whose support is necessary for nomination.
By contrast, PR allows candidates to target their message and appeal to ideological minorities and can fail to encourage them to push majoritarian messages. However, it also helps give a voice to a greater percentage of the electorate than do single-member districts, in which the electoral minority exists without any representation at all. PR can often encourage greater obedience to party hierarchies, on which candidates are dependent during the nomination and selection phases of a campaign, thereby allowing greater efficiency in governance.
https://www.fraserinstitute.org/studies/proportional-representation-in-practice-an-international-comparison-of-ballots-and-voting-rules