Problem: To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.
What ideas are used to understand groups beyond symbols?
---
Answer: structural concepts


Problem: Both the council members and the mayor serve four-year terms; none face term limits. Council members are nominated by their wards via a ward-level primary held in September. The top vote-earners from each party then compete at-large for their ward's seat on the November ballot. In other words, on election day the whole city votes on all the council races up for that year. Council elections are severed: Wards 1, 2, and 4 (as well as the mayor) are up for election in the same year (most recently 2011), while Wards 3, 5, and 6 share another year (most recently 2013).
Which wards elect city council members in the same year as the mayor?
---
Answer: 1, 2, and 4


Problem: The onset of the southwest monsoon is anticipated to reach Punjab by May, but since the early 1970s the weather pattern has been irregular. The spring monsoon has either skipped over the area or has caused it to rain so hard that floods have resulted. June and July are oppressively hot. Although official estimates rarely place the temperature above 46 °C, newspaper sources claim that it reaches 51 °C and regularly carry reports about people who have succumbed to the heat. Heat records were broken in Multan in June 1993, when the mercury was reported to have risen to 54 °C. In August the oppressive heat is punctuated by the rainy season, referred to as barsat, which brings relief in its wake. The hardest part of the summer is then over, but cooler weather does not come until late October.
When was the record heat spike in Multan?
---
Answer:
June 1993