Sets defined algebraically revision notes

Edexcel International GCSE Maths (4MA1), chapter 18: Sets, specification point 1.5a. Free lesson.

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Algebraically defined sets use a rule to decide which values from the universal set are members. Solve each rule, then compare the resulting sets to find intersections, unions and complements.

How to read and solve a set rule

The colon in set notation means such that. Everything after it is a condition: test the possible values in the universal set against that condition, then list the ones that fit. For ξ = {x : x is an integer, 1 ≤ x ≤ 10}, the possible values are 1 to 10. A = {x : 2x > 9} means x > 4.5, so A = {5, 6, 7, 8, 9, 10}. B = {x : x² < 30}, so B = {1, 2, 3, 4, 5}.

How to find an intersection, union and complement

The intersection A ∩ B contains values in both sets. The union A ∪ B contains values in either set, or both. A complement contains universal-set values outside the set. For the sets above, only 5 is in both A and B, so A ∩ B = {5}. Together A and B cover every value from 1 to 10, so A ∪ B is the whole universal set. Therefore, (A ∪ B)' has no members. This is the empty set, written ∅ or {}. Do not write {∅}: that means a set containing one member.

How to handle boundary values

A strict inequality, such as > or <, does not include a value on its boundary. Check the boundary by putting it back into the original rule, rather than deciding by appearance. For 2x > 9, x = 4 gives 2 × 4 = 8, which is not greater than 9. So 4 is not in A. The first integer in A is 5.

How equations with squares affect membership

A squared value can come from a positive or negative number, because squaring hides the sign. Check both solutions against the universal set and any other conditions. For ξ = {x : x is an integer, −3 ≤ x ≤ 3}, A = {x : x² = 4} gives A = {−2, 2}. B = {x : x + 1 > 0} means x > −1, so B = {0, 1, 2, 3}. Only 2 is in both sets, so A ∩ B = {2}. The values outside both sets are −3 and −1, so (A ∪ B)' = {−3, −1}.

When a set has no members

A set can have no values that satisfy its rule. This is not necessarily a mistake: the empty set is a valid answer, and its number of elements is zero. Write an empty set as ∅. If asked how many elements it contains, write n(A) = 0.

A quick check with multiples of four

Solve the inequality first, then keep only the values that also meet the multiple-of-four condition. For ξ = {x : x is an integer, 1 ≤ x ≤ 20}, P is the set of multiples of 4 and Q = {x : 3x > 30}. 3x > 30 gives x > 10. The multiples of 4 in the universal set that are greater than 10 are 12, 16 and 20.