Sets
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Set Notation and Venn Diagrams
A set is just a collection of things, and set notation gives a precise way to describe a set and to combine two sets. A Venn diagram shows this visually, sorting every element into the correct region.
- A set can be described by listing its elements, e.g. B = a, b, c, ..., or by a rule using set-builder notation, e.g. S = x : x is an integer, 1 ≤ x ≤ 20
- Key notation:
- n(A) - the number of elements in set A
- A' - the complement of A: every element in the universal set that is not in A
- E - the universal set: everything currently being considered
- A B - the union of A and B: everything in A, in B, or in both
- A B - the intersection of A and B: only the elements in both A and B
- A Venn diagram (Core level) shows the universal set as a rectangle, with up to two overlapping circles inside it for the sets being compared
Worked Example: Completing a Two-Set Venn Diagram
- Question: E = \x : x is an integer, 20 x 30\, A = multiples of 3, B = even numbers. Complete a Venn diagram for A and B, and find n(A B).
- Step 1: List the elements of A: A = \21, 24, 27, 30\
- Step 2: List the elements of B: B = \20, 22, 24, 26, 28, 30\
- Step 3: Numbers in both sets go in the overlap: A B = \24, 30\. Numbers only in A go in the A-only region: 21, 27. Numbers only in B go in the B-only region: 20, 22, 26, 28. The remaining numbers in 𝓔, 23, 25, 29, go outside both circles
- Step 4: A' is every element of 𝓔 that is not in A: A' = \20, 22, 23, 25, 26, 28, 29\
- Answer: n(A B) = 2
Every number from 20 to 30 placed in its correct region
A' is the shaded region: everything in 𝓔 that is outside circle A
Common Mistakes
MistakeConfusing union and intersection, e.g. treating A B as meaning only the shared elements
FixA B ("union") means everything in either set; A B ("intersection") means only the overlap - remember ∩ looks like the overlap of two circles
MistakeLeaving out elements of 𝓔 that belong to neither set when completing a Venn diagram
Fixevery element of the universal set must be placed somewhere - if it is not in A or B, it still goes inside the rectangle but outside both circles
Extended Set Notation and Three-Set Venn Diagrams
Extended Only
Extended level adds notation for individual elements and for one set sitting entirely inside another, and allows Venn diagrams to compare three sets at once instead of two.
- Further notation:
- x A - "x is an element of A"
- x A - "x is not an element of A"
- - the empty set: a set with no elements
- A B - "A is a subset of B": every element of A is also in B
- A B - "A is not a subset of B"
- A three-set Venn diagram uses three overlapping circles inside the universal set rectangle, giving up to eight regions in total
Worked Example: Using Set-Builder Notation with Three Sets
- Question: E = \x : x is an integer, 50 < x < 70\, P = square numbers, Q = odd numbers, R = multiples of 5. Find (i) P R, (ii) n(P Q R)'.
- Step 1: Within 𝓔, the only square number is P = \64\ (since 8² = 64), and the multiples of 5 are R = \55, 60, 65\
- Step 2: P and R share no elements, so P R =
- Step 3: P = 64 is even, so it cannot also be in Q (the odd numbers) - this means P Q is already empty, so P Q R = too
- Step 4: 𝓔 runs from 51 to 69, so n(E) = 19. Since n(P Q R) = 0, the complement contains every element of 𝓔
- Answer: (i) P R = ; (ii) n(P Q R)' = 19
Worked Example: Elements and Subsets
- Let A = 2, 3, 5, 7, 11. Since 7 is in the list, 7 A. Since 9 is not in the list, 9 A
- Let X = 2, 4, 6 and Y = 1, 2, 3, 4, 5, 6, 7, 8. Every element of X (2, 4 and 6) is also in Y, so X Y
X ⊆ Y: circle X sits entirely inside circle Y, since every element of X is also in Y
Common Mistakes
MistakeUsing ⊆ (subset) when relating a single element to a set, e.g. writing 3 \1, 2, 3\
Fix∈ relates a single element to a set (3 \1, 2, 3\); ⊆ relates one whole set to another whole set
MistakeThinking the empty set ∅ is the same as the set 0
Fix contains no elements at all, but 0 contains one element, the number zero - they are not the same set
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