0580 · 1.2Sets

Set notation and definitions

0:00Some students play football, others play basketball, and a few do both. Adding the two sports totals can count the same person twice. Sets give us a precise way to describe these groups and their overlap. By the end, you will understand set notation, interpret Venn diagrams, and organise information without counting anything twice. You only need a few starting ideas. You should recognise whole numbers, multiples, and simple inequalities.

0:24A multiple of three is a number such as three, six, or nine. An inequality describes a condition, such as x being greater than or equal to two. We will use these ideas to decide which objects belong to a set. A set is a well-defined collection of objects, called elements. Well-defined means we can decide whether a particular object belongs to the collection. The set A shown contains two, four, and six.

0:48Curly brackets enclose its elements, while commas separate them. Four is an element of A, but five is not. The membership symbol means is an element of. Its crossed version means is not an element of. These statements compare one object with a set. The notation n of A means the number of elements in A. Here, n of A equals three because the collection contains three different numbers, not because of their sum.

1:12Order never changes a set, and repeating an element does not create another member. Listing six, two, and four describes the same set as listing two, four, and six. Writing four twice would still represent just one element with that value. A set records which objects belong, rather than how often someone has written them. Sets do not always appear as complete lists. In a rule-based definition, read the colon as such that.

1:36The first example means the set of values x such that x is a natural number. The rule describes membership without listing every value. Always use any restrictions or number conventions supplied in the question. The second example contains ordered pairs, with x first and y second. Each pair belongs when y equals m times x plus c. Take m equal to two and c equal to one. The point with x equal to three and y equal to seven belongs, because seven equals two times three plus one.

2:04Here, each element is an entire coordinate pair, rather than an individual number. The third example contains values from a to b, including both endpoints. Unless another restriction is given, do not assume those values must be integers. The final example lists elements directly, with dots indicating continuation. Those letters represent elements, not an algebraic calculation. A continuation must be clear enough for membership to be understood. The universal set contains all the elements being considered in a particular question.

2:33Its symbol is the curly capital E shown beside the rectangle. It does not mean everything that exists. Here, our universe contains the integers from one to six. Every element in our diagram must therefore be one of those six numbers.

Venn diagrams and set operations

2:46A Venn diagram represents sets as regions inside this universal rectangle. An element goes inside a circle if it belongs to that set. The complement of A contains elements in the universal set that are not in A. Its notation is A with a small dash, read as A dash. Here, those elements are one, three, and five. The universal set is essential when finding a complement. Seven lies outside A, but it is also outside this universe.

3:13It therefore cannot belong to the complement of A in this question. If the universal set changes, the complement may change too. Always identify the universe before deciding what belongs outside a circle. A subset is a set whose every element belongs to another set. P contains two and six, so it is a subset of A. The subset symbol compares entire sets, rather than individual elements. Q contains two and three, so it is not a subset of A.

3:40The element three provides the reason because it does not belong to A. A set is also a subset of itself. Each of its elements certainly belongs to it, so equality is allowed by this subset symbol. Do not confuse membership with being a subset. Two is an element of A, while the set containing only two is a subset. The curly brackets change what we are describing. The union of A and B contains elements belonging to either set, or both.

4:05Shade all of both circles to represent it. The intersection of A and B contains elements belonging to both sets. Shade only their overlap to represent that collection. Union combines membership, while intersection requires the two conditions to hold together. The word or includes the overlap when describing a union. An element does not stop belonging to the union because it satisfies both conditions. For an intersection, however, satisfying only one condition is insufficient.

4:31Think of union as collecting everything from either group. Think of intersection as keeping only the shared members.

Worked example: two sets

4:37For our first worked example, the universe contains the integers from one to twelve. A contains the even numbers, giving two, four, six, eight, ten, and twelve. B contains multiples of three, giving three, six, nine, and twelve. Find their intersection, their union, the complement of A, and the number of elements in the union. Six and twelve appear in both lists, so they form the intersection. Combining the lists gives two, three, four, six, eight, nine, ten, and twelve.

5:05Those eight distinct numbers form the union. Removing A from the universe leaves one, three, five, seven, nine, and eleven. Therefore, n of the union equals eight. Place the shared elements in the overlap first. Six and twelve each appear once, because each is one element belonging to both sets. The remaining even numbers belong in A only. Three and nine belong in B only. The four numbers belonging to neither set go outside both circles, but remain inside the universal rectangle.

5:33More complicated notation combines the same simple conditions. The intersection of A with the complement of B means elements in A that are not in B. In our diagram, these are two, four, eight, and ten. Notice that being outside B does not necessarily mean being outside A. Translate each symbol into a membership condition before selecting a region. The complement of the union means elements belonging to neither set.

5:57Shade outside both circles, while staying inside the rectangle. In our example, that region holds one, five, seven, and eleven. A complement always applies to the complete expression inside its brackets. By contrast, the complement of the intersection means everything except the overlap. That includes both single-set regions as well as the region outside both circles. In our example, it holds ten numbers, which are all of them apart from six and twelve.

6:22Notice how moving the dash outside the brackets changes the shaded area completely. The empty set contains no elements, and its symbol is shown here. If two sets have no shared elements, their intersection is empty. Such sets are called disjoint, so their circles do not overlap. The number of elements in the empty set is zero. However, the set containing zero has one element, so it is not empty.

Worked example: counting students

6:46Our second worked example gives numbers of students rather than lists of elements. There are thirty students altogether, eighteen play football, and fourteen play basketball. Six play both sports, and these are included in each sport’s total. Find how many play football only, basketball only, and neither sport. We will start with the shared group. Put six in the overlap, since that many students play both sports. Subtract six from eighteen to obtain twelve who play football only.

7:12Subtract six from fourteen to obtain eight who play basketball only. Add twelve, six, and eight to obtain twenty-six who play at least one sport. Subtract twenty-six from thirty to obtain four who play neither. Adding eighteen and fourteen gives thirty-two, because the six shared students have been counted twice. Subtracting six once corrects that duplicate count, giving twenty-six. The complement of F includes everybody who does not play football.

7:35It therefore includes eight basketball-only students and four who play neither. 8 + 4 gives twelve students in the complement of F. Numbers written in these regions represent counts, not named elements. The twelve written in F only counts students, rather than naming one student. Read the question to establish which meaning is intended. Checking that all region counts total thirty confirms that the diagram accounts for the whole class.

7:59Each student should appear in exactly one region.

Three-set Venn diagrams and recap

8:02Our third worked example uses three sets within the integers from one to twelve. A holds the even numbers, B the multiples of three, and C the numbers less than or equal to six. We need a diagram representing all three conditions. Start in the central overlap, where an element must satisfy every condition at once. Six is even, a multiple of three, and within the limit for C, so it goes in the centre.

8:26Twelve belongs to A and B but fails the condition for C. It therefore goes in the overlap of A and B only. Two and four belong to A and C, but not B. Three belongs to B and C, but not A. These pair overlaps must exclude the central element already placed. Eight and ten belong to A only because neither satisfies the other conditions. Nine belongs to B only because it is odd and greater than six.

8:49One and five belong to C only because neither meets the other rules. Seven and eleven belong outside all three circles. Every integer from one to twelve now has exactly one position. An exam question could ask for the intersection of C with the union of A and B. We need elements inside C that also belong to at least one of the other sets. The qualifying elements are two, three, four, and six.

9:12Their set therefore contains four elements. One and five are excluded because belonging to C alone is insufficient. Common mistakes come from ignoring part of a condition. A union includes the overlap, and a complement stays within the universal set. A subset statement concerns every element, so one counterexample disproves it. In counting questions, a total for a whole circle includes its overlaps. Subtract those shared groups before writing a number in an only region.

9:39These methods must work without a calculator on Paper two, and they also apply on Paper four. The calculations here use exact whole-number counts, so rounding is unnecessary. When asked to list elements, use curly brackets rather than giving only their number. When asked to shade a region, translate the notation into conditions and check every part of the diagram. Sets describe membership, while Venn diagrams turn those membership conditions into visible regions.

10:03Union means either set or both, while intersection means the shared region only. A complement means outside the named set but inside the universe. A subset means every element of one set belongs to another. Understanding those conditions makes the symbols useful, rather than something to memorise.