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Algebra I – Terminology Sets and Expressions

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    Study Guide Algebra I – Terminology Sets and Expressions 1. Set Theory What Is a Set? A set is simply a collection of objects or numbers grouped together. For example: {1, 2, 3} This set contains the numbers 1, 2, and 3 . When we say 3 is part of the set , we write it using the symbol ∈ , which means “is an element of” or “belongs to.” Example : 3 ∈ {1,2,3} This reads as: “3 is an element of the set {1,2,3}.” Special Types of Sets Subsets A subset is a set that is completely contained inside another set. Example : {1,2} is a subset of {1,2,3} The larger set contains all the elements of the smaller set. There are two types of subsets: Proper Subset

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    Study Guide A proper subset is a subset that does not contain all elements of the original set. Example : {1,2} ⊂ {1,2,3} Here the symbol ⊂ means “proper subset of.” Improper Subset If a subset is exactly the same as the original set , it is called an improper subset . Example : {1,2,3} ⊆ {1,2,3} The symbol ⊆ means “subset of (possibly equal).” Universal Set The universal set is the complete set of all elements being considered in a particular situation. Think of it as the big group that contains everything we are studying . Empty Set (Null Set) The empty set (also called the null set ) is a set that contains no elements . It is written as: ∅ or { } Important note: The empty set is not written as { ∅ } , because that would mean a set containing one element (the empty set itself). Important Fact Both the universal set and the empty set are subsets of every set . Describing Sets There are two main ways to describe a set .

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    Study Guide 1. Rule Method In this method, we describe the elements of a set using a rule or condition . Example : {x : x > 3, x is a whole number} This means: All whole numbers greater than 3 So the set would be: {4, 5, 6, ...} Since the numbers continue forever, this type of set is often represented using a number line rather than listing all values. 2. Roster Method In the roster method , we simply list all the elements in the set . Example : {1,2,3} This set contains the numbers 1, 2, and 3 only . The same set can also be written using a rule. Example: {x : x < 4, x is a natural number} or {x : 0 < x < 4, x is a whole number} Both descriptions represent the set:

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    Study Guide {1,2,3} Incorrect Example {x : 0 < x < 4} This rule includes all numbers between 0 and 4 , such as: 1, 1.5, 2, 2.7, 3.2, etc. So it does not correctly describe the set {1,2,3}. Types of Sets Finite Sets A finite set has a limited (countable) number of elements . Example: {a, b, c, d, e} This set contains five elements , so it is a finite set . Infinite Sets An infinite set has unlimited elements . Example: {1,2,3,...} The numbers continue forever, so this is an infinite set . Comparing Sets Equal Sets

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    Study Guide Two sets are equal if they contain exactly the same elements , even if the order is different. Example: {1,2,3} = {3,2,1} The elements are the same, so the sets are equal. Equivalent Sets Two sets are equivalent if they have the same number of elements , even if the elements are different. Example: {1,2,3} and {a, b, c} Each set contains three elements , so they are equivalent. Venn Diagrams A Venn diagram (also called Euler circles ) is a visual way to show relationships between sets. Figure 1. A Venn diagram Explanation of the diagram:

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    Study Guide • A represents the elements inside the left circle . • B represents the elements inside the right circle . • C represents the overlapping area , which contains elements that belong to both A and B . Operations with Sets Union of Sets The union of two sets combines all elements from both sets . If an element appears in both sets, it is written only once . The symbol used for union is: ∪ Example: If A = {1,2,3} B = {3,4,5} Then A ∪ B = {1,2,3,4,5} Notice that 3 appears only once . The union of two sets combines all the elements from both sets into one set. The symbol used for union is: ∪ When forming the union, duplicate elements are written only once .

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    Study Guide Example 1 Find the union: {1,2,3} ∪ {3,4,5} Step 1: Write all elements from both sets. Set A → {1,2,3} Set B → {3,4,5} Step 2: Combine them and remove duplicates. {1,2,3} ∪ {3,4,5} = {1,2,3,4,5} Explanation: The number 3 appears in both sets , but it is written only once in the union. So, the union contains all the elements from both sets . Intersection of Sets The intersection of two sets contains only the elements that appear in both sets at the same time . The symbol used for intersection is: ∩ Example 2 Find the intersection: {1,2,3} ∩ {3,4,5} Step 1: Look for elements that appear in both sets . Set A → {1,2,3}

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    Study Guide Set B → {3,4,5} Step 2: Identify the common element. The number 3 appears in both sets . So, {1,2,3} ∩ {3,4,5} = {3} Explanation: The intersection contains only the shared element , which is 3 . Using Venn Diagrams to Understand Sets To better understand how sets relate to each other, we can use a Venn diagram . Figure 2 . Intersection of set A and set B Understanding the Diagram In the diagram: • Set A contains the numbers 1 and 2 .

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    Study Guide • Set B contains the numbers 4 and 5 . • The overlapping region contains 3 , which belongs to both sets . From this diagram: Union of A and B: A ∪ B = {1,2,3,4,5} Intersection of A and B: A ∩ B = {3} The union includes every number shown in the diagram , while the intersection includes only the overlapping element . Example 3 Find the intersection: {1,2,3} ∩ {4,5} Step 1: Check for elements that appear in both sets. Set A → {1,2,3} Set B → {4,5} Step 2: Compare the elements. There are no common elements in these two sets. So, {1,2,3} ∩ {4,5} = ∅ What Does ∅ Mean? The symbol ∅ represents the empty set (also called the null set ).

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    Study Guide It means that the sets do not share any elements . So, when two sets have no common members , their intersection is the empty set . 2. Quiz: Set Theory 1. Question Which of the following symbols represents “is an element of” ? Answer Choices • ∈ • ⊂ • ⊆ Correct Answer : ∈ Why This Is Correct The symbol ∈ means “is an element of” in set theory. It shows that a particular object belongs to a set. Example: 3 ∈ {1, 2, 3, 4} This means 3 is an element of the set {1,2,3,4}. The other symbols mean different things: • ⊂ means proper subset • ⊆ means subset (possibly equal)

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