Set Notation And Venn Diagrams
邊玩邊學
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課程筆記
Set Notation Basics
- A set is a collection of elements, written inside curly braces `{}`.
- `ℰ` denotes the universal set (all elements under consideration).
- `n(A)` is the number of elements in set A.
- `∈` means 'is an element of'; `∉` means 'is not an element of'.
- `∅` is the empty set (no elements).
- `A ⊆ B` means 'A is a subset of B' (all elements of A are in B).
- `A ⊈ B` means 'A is not a subset of B'.
Set Operations
- `A ∪ B` is the union of A and B (elements in A or B or both).
- `A ∩ B` is the intersection of A and B (elements in both A and B).
- `A'` is the complement of A (elements in ℰ not in A).
- `A ∪ B` includes all elements from both sets, counting overlaps only once.
- `A ∩ B` contains only the shared elements.
- `A'` depends on ℰ; e.g., if ℰ = {1,2,3,4,5} and A = {1,3}, then A' = {2,4,5}.
Intersection of A and B
Venn Diagrams – Structure
- A Venn diagram uses a rectangle for ℰ and circles for sets.
- Overlapping circles show intersections; non‑overlapping parts show elements unique to each set.
- The region inside a circle represents that set; outside the circle is its complement.
- Numbers or elements are placed in each region to show counts or members.
Sets A and B
Interpreting Venn Diagrams
- `A ∪ B` is the total shaded area inside either circle (including overlap).
- `A ∩ B` is the overlap region of the two circles.
- `A'` is everything outside the A circle (but inside ℰ).
- `(A ∪ B)'` is the region outside both circles (complement of union).
- `A ∩ B'` is the part of A that is not in B.
A intersect not-B
Problem‑Solving with Venn Diagrams
- Use given `n(A)`, `n(B)`, `n(A ∩ B)`, `n(ℰ)` to find missing region counts.
- For three sets, work from the centre (triple intersection) outward.
- Remember: total in ℰ = sum of all regions (including outside all sets).
- Form equations using `n(A ∪ B) = n(A) + n(B) – n(A ∩ B)`.
- Check that no region has a negative count; if so, adjust assumptions.
Three-set Venn diagram
Set Notation with Descriptions
- Sets can be defined by a rule: `{x : condition}` or `{x | condition}`.
- Example: `{x : x is an integer and 1 ≤ x ≤ 10}` means {1,2,…,10}.
- The colon or vertical bar is read as 'such that'.
- If no type is given, `x` can be any real number.
Common Exam Tips
- Always list elements without repetition in union problems.
- `n(A ∪ B)` includes elements in both sets only once.
- `A' ∩ B' = (A ∪ B)'` (De Morgan's Law).
- When shading, carefully identify the region described by the set expression.
- Double‑check that your Venn diagram satisfies all given totals.
投影片
練習題
免費預覽——45 題中的 8 題。註冊即可查看全部。
1.ℰ = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, A = {22, 24, 26, 28, 30}, B = {21, 24, 27, 30}. List the members of A ∩ B.
Easy- A{24, 30}
- B{22, 24, 26, 28, 30}
- C{21, 24, 27, 30}
- D{21, 22, 24, 27, 30}
2.ℰ = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, A = {22, 24, 26, 28, 30}. List the members of A'.
Easy- A{21, 23, 25, 27, 29}
- B{22, 24, 26, 28, 30}
- C{21, 22, 23, 24, 25, 26, 27, 28, 29, 30}
- D{23, 25, 27, 29}
3.ℰ = {letters of the alphabet}, B = {b, r, a, z, i, l}, I = {i, r, e, l, a, n, d}. List the members of B ∪ I.
Easy- A{b, r, a, z, i, l, e, n, d}
- B{b, r, a, z, i, l}
- C{i, r, e, l, a, n, d}
- D{a, i, l, r}
4.ℰ = {letters of the alphabet}, B = {b, r, a, z, i, l}, I = {i, r, e, l, a, n, d}. List the members of B ∩ I'.
Medium- A{b, z}
- B{a, i, l, r}
- C{e, n, d}
- D{b, r, a, z, i, l}
5.B = {b, l, u, e}, G = {g, r, e, y}, W = {w, h, i, t, e}. List all the members of B ∪ G.
Easy- A{b, l, u, e, g, r, y}
- B{b, l, u, e}
- C{g, r, e, y}
- D{e}
6.B = {b, l, u, e}, G = {g, r, e, y}, W = {w, h, i, t, e}. List all the members of W ∩ G'.
Medium- A{w, h, i, t}
- B{e}
- C{w, h, i, t, e}
- D{g, r, y}
7.Which of the following is the correct set notation for the empty set?
Easy- A∅
- Bℰ
- C0
- D{}
8.If ℰ = {1, 2, 3, 4, 5} and A = {1, 3}, what is A'?
Easy- A{2, 4, 5}
- B{1, 3}
- C{1, 2, 3, 4, 5}
- D{2, 4}
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