Set Notation Practice

1 Secondary 4 Mathematics: Sets 1. Definition of a Set • A set is a collection of distinct objects such as numbers, letters, symbols, etc. • The objects in a set are called the members or elements of the set. • Common examples of sets (a) Set of whole numbers, W = {0, 1, 2, 3, …) (b) Set of integers, Z = {…, –2, –1, 0, 1, 2, …} (c) Set of prime numbers, P = {2, 3, 5, 7, 11, …} 2. Ways of Defining a Set (a) A = {a, b, c} à A is the set containing elements a, b and c. (b) B = {x : x is a prime number} à B is the set containing set of prime numbers - B can also be expressed as B = {2, 3, 5, 7, 11, …} (c) C = {x : x is a integer number such that 5 ≤ x <9} - C can also be expressed as C = {5, 6, 7, 8} 3. Null or Empty Sets • A null or empty set, denoted by { } or ∅, is a set which contains no elements. 4. Equal Sets • Two sets are equal if they have the same elements. Example: Set A = {1, 2, 3} and set B = {3, 1, 2} Then A = B (Note that the order in which the elements appear in the set is not important).

2 5. Subset and Subset Symbols Symbol Meaning Example A ⊂ B Set A is a proper subset of set B. i.e., every element of A is also an element of B, and A ≠ B. Set A = {1, 2, 3} and set B = {3, 1, 2, 4, 5}, then A ⊂ B 𝐴 ⊆ 𝐵 A is subset of B i.e., every element of A is also an element of B, and set A can be equal set B. If A ⊆ B and Set B = {3,1, 2, 4, 5}, then Set A can be = {1, 2, 3} or Set A can be = {1, 2, 3, 4, 5} 𝐴 ⊄ 𝐵 A is not subset of B. If Set A = {1, 2, 5} and Set B = {1, 3, 6, 8}, then 𝐴 ⊄𝐵 𝐴 ∩ 𝐵 The intersection of sets A and B. It refers to the common elements found in both sets. If Set A = {1, 2, 3} and Set B = {1, 3, 6, 8}, then 𝐴 ∩ 𝐵 = {1, 3} 𝐴 ∪ 𝐵 The union of sets A and B. This refers to the set of elements which belongs to either set A or set B or both/ If Set A = {1, 2, 3} and Set B = {1, 3, 6, 8}, then 𝐴 ∪ 𝐵 = {1, 2, 3, 6, 8} n(A) Represents the number of elements in set A. If Set A = {1, 2, 3}, then n(A) = 3 6. Element of a Set • If a is an element of set A, we write a ∈ 𝐴. • If b is not an element of set A, we write b ∉ A. 7. Universal Set • A universal set is a set which contains all the available elements in a particular discussion. • It is the largest set in the discussion and is denoted by ξ.

3 8. Complement of Set • The complement of a set A, denoted by 𝐴′, relative to the universal set ξ contains elements that are not elements of A but are elements of ξ. i.e. it is the set that contains all elements in 𝛏 except those in A. Example: If ξ = {1, 2, 3, 4, 5}, 𝐴 = {1, 2}, then 𝐴′ = {3, 4, 5} 9. Disjoint Sets • Two sets are said to be disjoint if they do not have any common element. Example: If A = {1, 2, 3, 4, 5}, 𝐵 = {6, 7}, then 𝐴 and 𝐵 are disjoint. 10. Venn Diagram • A Venn diagram is pictorial representation of relationships involving sets. Example: If ξ = {1, 2, 3, 4, 5, 6, 7, 8}, 𝐴 = {1, 2, 3, 4}, 𝐵 = {2, 3, 6, 7}. Venn diagram of example: Venn diagram of (1) proper subset, (2) disjoint sets and (3) complement of a set: (1) Complement of 𝐴 (Shaded region) (2) 𝐴 and B are disjoint (3) B ⊂ 𝐴

4 Venn diagram representing (4) Intersection of sets and (5) Union of sets. Other Examples: Write down the set notation that represents the shaded region in the Venn diagram below: Answer: 𝑃′ ∪ 𝑄 Hint: Dissect the diagram into its individual parts: Venn diagram of 𝑃′ + Venn diagram of 𝑄 (4) 𝐴 ∩ 𝐵 (5) 𝐴 ∪ 𝐵 P Q P Q

5 Sets (Worksheet 1) 1. On each of the Venn diagram, shade the region(s) represented by the set notation (a) 𝐵′ ∩ 𝐴 (b) 𝐵′ ∩ 𝐴′ A B A B

6 (c) (𝐴′ ∩ 𝐵′)′ (d) 𝑃 ∪ 𝑄′ (e) (𝑃′ ∪ 𝑄)′ A B P Q P Q

7 2. It is given that ξ = {𝑥: 𝑥 is an integer such that 1 ≤ 𝑥 ≤ 15}, 𝐴 = {𝑥: 𝑥 integers divisble by 2} and 𝐵 = {𝑥: 𝑥 integers divisible by 3}. (a) List all the elements in 𝐴 ∪ 𝐵 in set notation. (b) List all the elements in 𝐴′ ∪ 𝐵 in set notation. (c) List all the elements in 𝐴 ∩ 𝐵 in set notation. (d) List all the elements in 𝐴′ ∩ 𝐵′ in set notation.

8 3. It is given that ξ = {𝑥: 𝑥 is an integer such that 1 ≤ 𝑥 ≤ 10}, 𝐴 = { 𝑥: 𝑥 is an odd number} and 𝐵 = {𝑥: 𝑥 integers multiple of 3}. (a) List all the elements in 𝐴 ∩ 𝐵 in set notation. (b) List all the elements in 𝐴′ ∩ 𝐵′ in set notation. (c) Draw a venn diagram showing ξ, 𝐴 and 𝐵 and place each of the members/elements in appropriate parts of the diagram. ξ

9 4. There are 27 children in a class. Out of these children, 19 own a smartphone, 15 own a computer and 3 neither own a smartphone no a computer. x number of student own both a smartphone and computer. Using a Venn diagram, find the number of children who own a smartphone but not a computer. Answer: ________________

10 5. It is given that ξ = {𝑥: 𝑥 is an integer such that 10 ≤ 𝑥 ≤ 23}, 𝐸 = {𝑥: 𝑥 is an even number}, 𝑃 = {𝑥: 𝑥 is a prime number}, and 𝑀 = {𝑥: 𝑥 is multiple of 5}, The Venn diagram shows the universal set ξ and the subsets 𝐸 and 𝑀. Two elements of ξ are shown in their appropriate subsets. (a) Complete the Venn diagram shown above and draw subset P. Write the remaining elements of ξ in the appropriate subsets of your Venn diagram. (b) Hence, simplify (𝐸′ ∪ 𝑀) ∩ 𝑃′ Answer: (b) (𝐸′ ∪ 𝑀) ∩ 𝑃′ = ________________ ξ E M 22 20