📘intermediate algebra review

key term - ∪

Definition

The union symbol, ∪, represents the operation of combining two or more sets into a single set that contains all the unique elements from the original sets. It is a fundamental concept in set theory and is particularly relevant in the context of solving compound inequalities.

5 Must Know Facts For Your Next Test

  1. The union of two sets A and B, denoted as A ∪ B, is the set that contains all the elements that are in either A, B, or both.
  2. The union operation is commutative, meaning that A ∪ B = B ∪ A.
  3. The union operation is associative, allowing for the combination of more than two sets, such as A ∪ B ∪ C.
  4. When solving compound inequalities, the union symbol is used to represent the 'or' condition, where the solution set includes all values that satisfy at least one of the individual inequalities.
  5. The union of two or more intervals on the number line can be represented using the union symbol, which is particularly useful in visualizing and understanding the solution set of a compound inequality.

Review Questions

  • Explain how the union operation is used in the context of solving compound inequalities.
    • When solving compound inequalities, the union symbol, ∪, is used to represent the 'or' condition. This means that the solution set includes all values that satisfy at least one of the individual inequalities. For example, the compound inequality x < -2 ∪ x > 3 has a solution set that includes all values less than -2 or greater than 3. The union operation allows us to combine these two separate solution sets into a single set that satisfies the overall compound inequality.
  • Describe the properties of the union operation and how they relate to set theory and compound inequalities.
    • The union operation has two important properties: commutativity and associativity. Commutativity means that the order of the sets does not matter, so A ∪ B = B ∪ A. Associativity allows for the combination of more than two sets, such as A ∪ B ∪ C. These properties are useful in set theory and in the context of solving compound inequalities, as they allow for the flexible manipulation and representation of the solution sets. For example, the compound inequality x < -2 ∪ x > 3 ∪ x = 1 can be simplified to x < -2 ∪ x > 1, taking advantage of the associative property of the union operation.
  • Analyze how the visual representation of the union operation on the number line can aid in understanding the solution set of a compound inequality.
    • The union of two or more intervals on the number line can be represented using the union symbol, ∪. This visual representation is particularly useful in understanding the solution set of a compound inequality. By plotting the individual inequalities on the number line and then taking the union of the resulting intervals, you can clearly see the complete solution set that satisfies the overall compound inequality. This approach helps to develop a deeper understanding of the relationship between the union operation, set theory, and the solving of compound inequalities, as the visual representation allows you to intuitively grasp the logic and structure of the problem.