Set Notation And Venn Diagrams

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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.

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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}

Thêm câu hỏi

9.If n(A) = 5 and A = {a, b, c, d, e}, which statement is correct?
Easy
- An(A) = 5
- Bn(A) = 4
- Cn(A) = 6
- Dn(A) = 0
10.Which symbol means 'is an element of'?
Easy
- A∈
- B∉
- C⊆
- D∩
11.There are 32 students in a class. 5 do not study any languages. 15 study German (G). 18 study Spanish (S). How many students study both German and Spanish?
Medium
- A6
- B4
- C5
- D8
12.ℰ = {x : 41 ≤ x ≤ 50}, A = {x : x is odd}, B = {x : x is a multiple of 3}, C = {x : x is prime}. List the elements of A ∩ C.
Medium
- A{41, 43, 47}
- B{41, 43, 47, 49}
- C{43, 47}
- D{41, 43, 47, 53}
13.ℰ = {x : 41 ≤ x ≤ 50}, A = {x : x is odd}, B = {x : x is a multiple of 3}, C = {x : x is prime}. Find n(A ∩ B ∩ C).
Medium
- A0
- B1
- C2
- D3
14.For sets C and D, what does C ∩ D′ represent?
Medium
- AThe part of C that is not in D
- BThe part of D that is not in C
- CThe intersection of C and D
- DThe union of C and D
15.n(ℰ) = 20, n(A ∪ B)' = 3, n(A) = 10, n(B) = 13. Find n(A ∩ B).
Hard
- A6
- B7
- C5
- D4
16.ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. Which of the following is A'?
Easy
- A{1, 3, 5, 7, 9}
- B{2, 4, 6, 8, 10}
- C{1, 2, 3, 4, 5}
- D{6, 7, 8, 9, 10}
17.If ℰ = {x : x is an integer and 1 ≤ x ≤ 12}, A = {x : x is a factor of 12}, B = {x : x is a multiple of 3}. Which of the following is A ∩ B?
Medium
- A{3, 6, 12}
- B{3, 6, 9, 12}
- C{1, 2, 3, 4, 6, 12}
- D{6, 12}
18.ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}. Which of the following is A ∪ B?
Easy
- A{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- B{1, 3, 5, 7, 9}
- C{2, 4, 6, 8, 10}
- D∅
19.ℰ = {a, b, c, d, e, f, g, h}. A = {a, b, c, d}, B = {c, d, e, f}. Which of the following is (A ∩ B)'?
Medium
- A{a, b, e, f, g, h}
- B{a, b, c, d, e, f}
- C{g, h}
- D{c, d}
20.Given n(ℰ) = 30, n(A) = 15, n(B) = 12, n(A ∩ B) = 5. What is n(A ∪ B)?
Medium
- A22
- B27
- C20
- D17
21.ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}. Which of the following is A' ∩ B'?
Medium
- A∅
- B{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- C{2, 4, 6, 8, 10}
- D{1, 3, 5, 7, 9}
22.n(ℰ) = 50, n(A ∪ B) = 40, n(A) = 25, n(B) = 30. What is n(A ∩ B)?
Medium
- A15
- B10
- C20
- D5
23.ℰ = {x : x is an integer and 1 ≤ x ≤ 15}. A = {x : x is a prime number}, B = {x : x is a multiple of 4}. Which of the following is A ∩ B?
Medium
- A∅
- B{2}
- C{4, 8, 12}
- D{2, 4, 8, 12}
24.If A = {a, b, c} and B = {a, b, c, d}, which of the following is true?
Easy
- AA ⊆ B
- BB ⊆ A
- CA = B
- DA ⊈ B
25.Which symbol represents the empty set?
Easy
- A∅
- Bℰ
- C∈
- D⊆
26.ℰ = {students in a class}. A = {students who play tennis}, B = {students who play cricket}. There are 30 students. 15 play tennis, 10 play cricket, and 5 play both. How many students play neither tennis nor cricket?
Medium
- A10
- B5
- C15
- D20
27.n(ℰ) = 40, n(A ∩ B) = 8, n(A ∪ B) = 25, n(A) = 15. What is n(B)?
Medium
- A18
- B10
- C20
- D22
28.ℰ = {1, 2, 3, 4, 5, 6, 7, 8}. A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. Which of the following is A ∩ B'?
Medium
- A{1, 2}
- B{3, 4}
- C{5, 6}
- D{1, 2, 3, 4}
29.n(ℰ) = 60, n(A) = 25, n(B) = 30, n(A ∩ B) = 10. What is n(A' ∩ B')?
Medium
- A15
- B20
- C25
- D10
30.If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, which of the following is true?
Easy
- AA ⊆ B
- BB ⊆ A
- CA = B
- DA ⊈ B
31.ℰ = {x : x is an integer and 1 ≤ x ≤ 10}. A = {x : x is odd}, B = {x : x is a factor of 10}. Which of the following is A ∪ B?
Medium
- A{1, 2, 3, 5, 7, 9, 10}
- B{1, 3, 5, 7, 9}
- C{1, 2, 5, 10}
- D{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
32.The symbol ∪ means 'intersection'.
Easy
True or false?
33.The universal set is always denoted by ℰ.
Easy
True or false?
34.If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∩ B?
Easy
- A{2, 3}
- B{1, 4}
- C{1, 2, 3, 4}
- D∅
35.In a Venn diagram, which region represents A ∪ B?
Easy
- AThe region inside either A or B or both
- BThe region inside both A and B
- CThe region outside both A and B
- DThe region inside A only
36.If ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, which of the following is A'?
Easy
- A{1, 3, 5, 7, 9}
- B{2, 4, 6, 8, 10}
- C{1, 2, 3, 4, 5}
- D∅
37.Match each set notation symbol with its meaning.
Easy
- ∈
- ∉
- ⊆
- ∪
- is an element of
- is not an element of
- is a subset of
- union
38.Arrange the following steps in order to find n(A ∪ B) when n(ℰ)=30, n(A)=15, n(B)=12, n(A∩B)=5.
Medium
- Use formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- Substitute numbers: 15 + 12 - 5
- Calculate: 22
- Identify n(A), n(B), n(A∩B)
39.Given ℰ = {1,2,3,4,5,6,7,8,9,10}, A = {2,4,6,8,10}, B = {1,2,3,4,5}. Which of the following are elements of A ∩ B? (Select all that apply)
Medium
- A2
- B4
- C6
- D8
- E1
40.The symbol ∅ represents the empty set.
Easy
True or false?
41.Given n(ℰ)=50, n(A)=30, n(B)=25, n(A∩B)=10. What is n(A' ∩ B')?
Medium
- A5
- B10
- C15
- D20
42.Fill in the blank with the correct set notation symbol.
Easy
The symbol for the complement of set A is ____.
43.In a class of 30 students, 18 play football, 15 play basketball, and 8 play both. How many students play neither sport?
Medium
44.Write down the set notation for the complement of A.
Easy
45.In a class of 30 students, 18 play football, 15 play basketball, and 8 play both. How many students play neither sport?
Medium
- A6
- B5
- C4
- D7

Ghi chú ôn tập

## Set Notation & Venn Diagrams ### Key Definitions - **Set**: collection of elements (numbers, letters, etc.) written in curly brackets `{}`. - **Universal set (ℰ)**: everything under consideration. - **Empty set (∅)**: set with no elements. - **n(A)**: number of elements in set A. ### Set Notation - `∈` : element of (e.g., 3 ∈ {1,2,3}) - `∉` : not an element of - `A ⊆ B` : A is a subset of B (all elements of A are in B) - `A ∩ B` : intersection (elements in both A and B) - `A ∪ B` : union (elements in A or B or both) - `A'` : complement (elements in ℰ not in A) ### Venn Diagrams - Rectangle = ℰ, circles = sets. - Overlap = intersection. - Outside circles = complement. ### Worked Example ℰ = {1,2,3,4,5,6,7,8}, A = {2,6,12,14,28}, B = {7,14,21,28,35} - n(A) = 5 - {14,28} = A ∩ B - A ∪ B = {2,6,7,12,14,21,28,35} - A' ∩ B' = {1,5,8} (not empty) ### Tips - For union, list each element once. - Complement is everything outside the set. - Use set notation to describe regions (e.g., `M ∩ D'`).

Set: A collection of elements, written in curly brackets {}.
Universal set (ℰ): The set of all elements under consideration.
Empty set (∅): The set with no elements.
n(A): The number of elements in set A.
∈: Is an element of.
∉: Is not an element of.
A ⊆ B: A is a subset of B; all elements of A are also in B.
A ∩ B: Intersection of A and B; elements in both sets.
A ∪ B: Union of A and B; elements in A or B or both.
A': Complement of A; all elements in ℰ not in A.
Venn diagram: A diagram with a rectangle (ℰ) and circles (sets) showing relationships.

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