Sets and set notation: worked examples and method

A set

A set is a collection of things in curly brackets, such as {2, 4, 6, 8}. Each member is written once and the order does not matter. The symbol ∈ means is a member of, so 4 ∈ {2, 4, 6}.

The symbols

ξ is the universal set, everything in the question. A ∪ B is in either, A ∩ B is in both, A' is not in A, and n(A) is how many members A has.

List a set

List the members that fit the words. With ξ as the numbers 1 to 12, the multiples of 3 are {3, 6, 9, 12} and the factors of 12 are {1, 2, 3, 4, 6, 12}. The number 1 is not prime.

Union and intersection

List both sets. The intersection keeps members that appear in both lists. The union joins the lists and writes each member once.

Complement

A' is everything in ξ that is not in A. List ξ first, cross out the members of A, and the rest is A'. Then n(A) + n(A') = n(ξ).

Count them

n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Take away the members counted twice. A rule such as {x : x > 3} means the set of all x such that x is greater than 3.

What the symbols mean

Curly brackets hold the members of a set, separated by commas. The order does not matter and a member is never listed twice, so \3, 1, 2\ and \1, 2, 3\ are the same set. The symbol in says that something is a member, so 4 in \2, 4, 6\ is true, and the symbol notin says it is not, so 5 notin \2, 4, 6\ is true. A set with no members is the empty set, written varnothing or \ \.

The universal set xi contains every item the question is talking about, for example the whole numbers from 1 to 12. Every other set in the question is made from members of xi. When one set sits entirely inside another, we write A subset B. For example, if A = \2, 4\ and B = \1, 2, 3, 4\ then A subset B, because 2 and 4 are both in B. Use in for a single member and subset for a whole set.

Method: 1. Read each symbol and say it in words before you do anything else. 2. Write the universal set out in full so that you can see every possible member. 3. Use the symbol in for one member and subset for a set inside another set.

Tip: Say the symbol out loud: the cap is for both, the cup is for either.

Listing the members of a set

Questions often describe a set in words, and you list the members. With xi as the whole numbers from 1 to 12, the multiples of 3 are \3, 6, 9, 12\ and the factors of 12 are \1, 2, 3, 4, 6, 12\. The even numbers are \2, 4, 6, 8, 10, 12\ and the square numbers are \1, 4, 9\. Only list members that are in xi, so the multiples of 3 stop at 12.

Check the definitions carefully, because they decide the edges. A factor list always includes 1 and the number itself. The number 1 is not a prime number, because a prime has exactly two factors, so the primes up to 12 are \2, 3, 5, 7, 11\. Counting the members of a set is then easy: n(primes) = 5. Write every list in order, and count it once it is finished.

Method: 1. Write the universal set out in full. 2. Go through it in order and pick out the numbers that fit the description. 3. Check the edges, such as whether 1 is prime and whether the number itself is a factor. 4. Count the members if the question asks for n of the set.

Tip: The number 1 is a factor of everything but it is not prime.

Union, intersection and complement

To combine two sets, list them both and then apply the symbol. Take xi = \1, 2, 3, dots, 12\ with A = \2, 4, 6, 8, 10, 12\ and B = \3, 6, 9, 12\. The intersection A cap B keeps the members in both lists, which is \6, 12\. The union A cup B puts everything from either list together, with each member written once, which is \2, 3, 4, 6, 8, 9, 10, 12\.

The complement A' is everything in xi that is not in A, so A' = \1, 3, 5, 7, 9, 11\. Counting gives n(A) = 6, n(B) = 4 and n(A cap B) = 2. Then n(A cup B) = 6 + 4 - 2 = 8, which matches the list of eight members. The subtraction is there because the members 6 and 12 were counted in both A and B. You can always check a count by listing the members.

Method: 1. List xi, A and B in full. 2. For A cap B keep only the members that appear in both lists. 3. For A cup B join the lists and write each member once. 4. For A' list the members of xi that are not in A, then check n(A) + n(A') = n(xi).

Describing a set with a rule

A set can be described by a rule instead of a list. The notation \x : x > 3\ is read as the set of all x such that x is greater than 3. The colon means such that. If the question names the type of number, you can list the members. The set \x : x is an integer, -2 < x ≤ 3\ contains the integers bigger than -2 and no bigger than 3, so it is \-1, 0, 1, 2, 3\ with 5 members.

Look at the inequality signs. A strict sign such as < leaves the end number out, and the sign ≤ keeps it in. The number -2 is not in the set above, and 3 is. If the rule says the numbers are not integers, there are too many to list, and the set is described by the rule instead. Rules and lists can be mixed in a question, so translate the rule into a list first.

Method: 1. Read the rule in words: the set of all x such that. 2. Decide which type of number is allowed, such as integers or prime numbers. 3. Use the inequality signs to decide which end numbers are in and which are out. 4. List the members in order.