Numbers & Operations
Sets
A set is the mathematical name given to a group of items that share some common property. All positive numbers make up one set, and all prime numbers make up another. Each item in a set is called an element or a member.
Don’t confuse a set with a sequence. A set is simply a collection of elements that are not necessarily related to one another, as they are in a sequence.
The union of two sets is another set that contains all the elements of each set. If set A contains all the blue-eyed women and set B contains all the blue-eyed men, the union of sets A and B is all blue-eyed women and men. If set A = (1, 2, 4, 6, 8) and set B = (2, 3, 5, 7, 8), the union of A and B is (1, 2, 2, 3, 4, 5, 6, 7, 8, 8).
The intersection of two sets is another set that contains all the elements the two sets have in common. If set A = (1, 2, 4, 6, 8) and set B = (2, 3, 5, 7, 8), the intersection of A and B is the set (2, 8).
A difficult set item involves a group of people, some of whom are engaged in activity A and others in activity B, while still others refrain from participating in either activity. Here’s an example:
7. In a group of students, 24 are considered to be good at math, 14 are good at liberal arts, and 12 are good at both. How many students are in the group?
(A) 14
(B) 24
(C) 26
(D) 38
(E) 50
To solve this item, use the following simple formula:
Total in a set = number in set 1 + number in set 2 – intersection of set 1 and set 2 + number in neither set
You have to figure out which members belong to set 1, which to set 2, what the intersection of two sets is, and how many abstain from participating.
Total students = those in math (24) + those in liberal arts (14) – those in both (12) + those in neither (0) = 26.
There are a total of 26 students, choice C.
Some set items may not explicitly tell you how many people are in neither set. If an item says that each student in a class has to learn either French or Italian, then the “neither set” (those learning neither French nor Italian) is zero.
That covers the numbers & operations basics. Now let’s apply these concepts to some SAT strategies.
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