Problem 39 - Algebra and Trigonometry

Question

\(39-40=\) Find the indicated set if \(\begin{array}{cc}{A=\\{x | x \geq-2\\}} & {B=\\{x | x<4\\}} \\ {C=\\{x |-1

Step-by-Step Solution

Answer

(a) \(B \cup C = \{x | -1 < x < 5\}\); (b) \(B \cap C = \{x | -1 < x < 4\}\).

Step 1: Understanding Sets

We need to find the union and intersection of sets \(B\) and \(C\). Set \(B\) is defined as \(\{x | x < 4\}\) and set \(C\) is defined as \(\{x | -1 < x \leq 5 \}\). When finding the union \(B \cup C\), we include all elements present in either set. For the intersection \(B \cap C\), we include only the elements present in both sets.

Step 2: Finding the Union \(B \cup C\)

The union \(B \cup C\) includes all elements from both \(B\) and \(C\). Since \(B = \{x | x < 4\}\) and \(C = \{x | -1 < x \leq 5 \}\), the union will include all values from \(-1 < x < 5\). Thus, \(B \cup C = \{x | -1 < x < 5\}\).

Step 3: Finding the Intersection \(B \cap C\)

The intersection \(B \cap C\) includes only the elements found in both sets. Since \(B = \{x | x < 4\}\) and \(C = \{x | -1 < x \leq 5 \}\), the intersection will be all overlaps, which are values from \(-1 < x < 4\). Thus, \(B \cap C = \{x | -1 < x < 4\}\).

Key Concepts

Union of Sets

The union of two sets, denoted as \(B \cup C\), combines all elements from both sets without duplicating any elements. In simple terms, it's like mixing all items from two boxes without worrying about duplicates.
For sets \(B\) and \(C\) given in the problem:

• Set \(B = \{x \mid x < 4\}\)
• Set \(C = \{x \mid -1 < x \leq 5\}\)

When finding the union \(B \cup C\), include all numbers in both sets' ranges. Therefore:- From Set \(B\), include numbers less than 4.- From Set \(C\), include numbers greater than -1 and up to 5 inclusively.Thus, the combined range from Set \(B\) and Set \(C\) in this problem becomes \(\{-1 < x < 5\}\). This means that any number greater than -1 but less than 5 is part of the union \(B \cup C\).
In essence, the union combines all possibilities from both sets into one larger set, making sure we don't repeat elements.

Intersection of Sets

The intersection of two sets, denoted as \(B \cap C\), includes only the elements that are present in both sets simultaneously. Think of it like finding common friends in two different social circles.
Let's look at the sets from the problem again:

• Set \(B = \{x \mid x < 4\}\)
• Set \(C = \{x \mid -1 < x \leq 5\}\)

For the intersection \(B \cap C\), we need to identify the overlapping range:- Numbers from Set \(B\) are anything less than 4.- Numbers from Set \(C\) are those greater than -1 and not exceeding 5.The common numbers between these two sets are actually those greater than -1 but still less than 4, since that's the overlap where both sets agree. Thus, \(B \cap C = \{-1 < x < 4\}\).
The concept of intersection helps highlight the shared elements, providing a focused view into what two sets have in common.

Inequality Notation

Inequality notation is a way to describe a range of values that a variable can take. It is very helpful in defining set boundaries in mathematical expressions.
In this problem, we use inequality notation to describe the elements of sets \(B\) and \(C\):

• \(B = \{x \mid x < 4\}\) means "all \(x\) such that \(x\) is less than 4."
• \(C = \{x \mid -1 < x \leq 5\}\) means "all \(x\) such that \(x\) is greater than -1 and less than or equal to 5."

In inequality notation, the symbols "\(<\)", "\(>\)", "\(\leq\)", and "\(\geq\)" are used:- \(<\) and \(>\) indicate less than and greater than, respectively.- \(\leq\) and \(\geq\) denote less than or equal to and greater than or equal to.This notation is precise, allowing clear communication of which values are included in a set. Understanding and applying inequality notation are foundational skills in mastering set theory and other mathematical concepts.