Sets PDF

CHAPTER 2 ⃒ SETS ______________________________________________________________________________________________ In mathematics, a set is a collection of distinct objects. The word “distinct” means that the objects of the set must be all different. For example, the set of counting numbers and...

CHAPTER 2 ⃒ SETS ______________________________________________________________________________________________ In mathematics, a set is a collection of distinct objects. The word “distinct” means that the objects of the set must be all different. For example, the set of counting numbers and the set of vowels. The objects that belong in a set are the elements or members of a set. A set is usually denoted by capital letters and elements are denoted by small letters. If 𝑎 is an element of a set 𝐴, then we say 𝑎 ∈ 𝐴 (𝑎 belongs to 𝐴). If 𝑎 is not an element of set 𝐴, then we say 𝑎 ∉ 𝐴 (𝑎 does not belong to 𝐴). An empty set or null set, denoted by ∅ or { }, is the set that has no elements. For example, the set of all green carabaos. There are no such carabaos that are color green, thus, it is empty or null. Three methods of describing sets are rule method, roster method, and set-builder notation. The rule method describes the set using words, the roster method lists the elements of the set inside a pair of braces with the elements separated by commas, and the set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy. Example 2.1. Let the sets 𝐴 = The set of distinct letters of the word "mathematics" 𝐵 = The set of all counting numbers that are multiples of 3 𝐶 = The set of natural numbers greater than 10. Then write the set 𝐴 in roster method, set 𝐵 in rule method, and set 𝐶 in set-builder notation. Solution 1. 𝐴 = {𝑚, 𝑎, 𝑡, ℎ, 𝑒, 𝑖, 𝑐, 𝑠} 2. 𝐵 = {𝑦 | 𝑦 = 3𝑥 for some integers 𝑥} 3. 𝐶 = {𝑥 ∈ ℕ | 𝑥 > 10} A set is finite if the number of elements in the set is a whole number. The cardinal number of a finite set is the number of elements in the set. The cardinal number of a finite set A is denoted by 𝑛(𝐴). For example, 𝐴 = {1, 2, 3, 4, 5}, then 𝑛(𝐴) = 5. This is read as “𝐴 has a cardinality of 5.” Set 𝐴 is equal to set 𝐵, denoted by 𝐴 = 𝐵, if and only if 𝐴 and 𝐵 have exactly the same elements. For example, let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝐵 = {𝑑, 𝑎, 𝑐, 𝑏}. Then 𝐴 = 𝐵. Set 𝐴 is equivalent to set 𝐵, denoted by 𝐴~𝐵, if and only if 𝐴 and 𝐵 have the same number of elements. For example, let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒} and 𝐵 = {2, 3, 5, 7, 9}. Then 𝐴~𝐵.  Sets and Logic ______________________________________________________________________________________________ Complements, Subsets and Venn Diagrams The set of all elements that are being considered is called the universal set denoted by 𝑈. Set 𝐴 is a subset of set 𝐵, denoted by 𝐴 ⊆ 𝐵, if and only if every element of 𝐴 is also an element of 𝐵. Clearly, 𝐴 ⊆ 𝐴 and ∅ ⊆ 𝐴, for any set 𝐴. A set with 𝑛 elements has 2 𝑛 subsets. For example, let 𝐴 = {𝑎, 3, 𝑦}. Then the subsets of 𝐴 are ∅, {𝑎}, {3}, {𝑦}, {𝑎, 3}, {𝑎, 𝑦}, {3, 𝑦}, and {𝑎, 3, 𝑦}. Since 𝐴 has 3 elements, so the number of subsets of 𝐴 is 8. A diagram showing the relationship among sets and between elements and sets is called a Venn diagram, named after the English logician James Venn (1834 – 1883). In a Venn diagram, the universal set 𝑈 is usually represented by a rectangle, while the other sets, which are subsets of 𝑈, are usually represented by circles, triangles, and other closed geometric figures. Operations on sets 1. The union of two sets 𝐴 and 𝐵, denoted by 𝐴 ∪ 𝐵, is the set of all elements that belong to either 𝐴, or 𝐵, or both 𝐴 and 𝐵. In set descriptive notation we have 𝐴 ∪ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}. Figure 2.1. Venn Diagram of 𝐴 ∪ 𝐵 In Figure 2.1, the shaded regions represent the set 𝐴 ∪ 𝐵. 2. The intersection of two sets 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵, is the set of all elements that belong to both sets 𝐴 and 𝐵. In set descriptive notation, we have 𝐴 ∩ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}.  Sets and Logic ______________________________________________________________________________________________ Figure 2.2. Venn diagram of 𝐴 ∩ 𝐵 In Figure 2.2, the shaded region represents the set 𝐴 ∩ 𝐵. 3. The complement of a set 𝐴, denoted by 𝐴’, is the set of all elements of the universal set 𝑈 that are not elements of 𝐴. In set descriptive notation, we have 𝐴′ = {𝑥 ∈ 𝑈: 𝑥 ∉ 𝐴}. Figure 2.3. Venn diagram of 𝐴′. In Figure 2.3, the shaded region represents the set 𝐴’. 4. The set difference (or the complement of set 𝐵 relative to set 𝐴) is the set of all elements that are in 𝐴 and not in 𝐵. In set descriptive notation, we have 𝐴\𝐵 = 𝐴 ∪ 𝐵′ = {𝑥 ∶ 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵}.  Sets and Logic ______________________________________________________________________________________________ Figure 2.4. Venn diagram of 𝐴\𝐵 In Figure 2.4, the shaded region represents 𝐴\𝐵. 5. A symmetric difference of two sets 𝐴 and 𝐵, denoted by 𝐴 Δ 𝐵, is the set of elements which are in either of the sets 𝐴 and 𝐵 and not in their intersection. In the set descriptive notation, we have 𝐴 Δ 𝐵 = (𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵) = {𝑥: 𝑥 ∈ (𝐴 ∪ 𝐵) 𝑎𝑛𝑑 𝑥 ∉ (𝐴 ∩ 𝐵)} Figure 2.5 The Venn diagram of 𝐴 Δ 𝐵 In Figure 2.5, the shaded region represents 𝐴 Δ 𝐵. 6. The Cartesian product of two sets 𝐴 and 𝐵, denoted by 𝐴 × 𝐵, is the set of all ordered pairs (a, b) where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. In the set descriptive notation, we have 𝐴 × 𝐵 = {(𝑎, 𝑏): 𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵}.  Sets and Logic ______________________________________________________________________________________________ Example 2.2 Let 𝑈 = {𝑥: 𝑥 ∈ ℤ, 0 < 𝑥 ≤ 10} 𝐴 = {𝑥: 𝑥 is an odd number less than 10 divisible by 3} 𝐵 = {𝑥: 𝑥 is an even number less than 5} 𝐶 = {𝑥: 𝑥 ∈ ℤ, 0 < 𝑥 < 6}. Then find 𝐴 ∪ 𝐶, 𝐴 ∩ 𝐵, 𝐵 ∩ 𝐶, 𝐵′, 𝐶\𝐴, 𝐴 × 𝐵. Solution Note that 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 𝐴 = {3, 9}, 𝐵 = {2, 4}, 𝐶 = {1, 2, 3, 4, 5}. We have 𝐴 ∪ 𝐶 = {3, 9} ∪ {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 9} 𝐴 ∩ 𝐵 = {3, 9} ∩ {2, 4} = ∅ 𝐵 ∩ 𝐶 = {2, 4} ∩ {1, 2, 3, 4, 5} = {2, 4} 𝐵′ = {2, 4}′ = {1, 3, 5, 6, 7, 8, 9, 10} 𝐶\𝐴 = {1, 2, 3, 4, 5}\{3, 9} = {1, 2, 4, 5} 𝐴 × 𝐵 = {3, 9} × {2, 4} = {(3, 2), (3, 4), (9, 2), (9, 4)} Properties of Set Operations 1. Commutative Properties a) 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 b) 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 2. Associative Properties a) (𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶) b) (𝐴 ∩ 𝐵) ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶) 3. Distributive Properties a) (𝐴 ∪ 𝐵) ∩ 𝐶 = (𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶) b) (𝐴 ∩ 𝐵) ∪ 𝐶 = (𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶) 4. De Morgan’s Laws a) (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′ b) (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵′