Problem 43 - Precalculus Mathematics for Calculus

Question

Find the indicated set if $$\begin{array}{c}A=\\{x | x \geq-2\\} \quad B=\\{x | x<4\\} \\\C=\\{x |-1

Step-by-Step Solution

Answer

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

Step 1: Determine Set B

Set B is defined as the set of all numbers where \( x < 4 \). Therefore, set B is expressed as \( B = \{ x | x < 4 \} \).

Step 2: Determine Set C

Set C is defined as the set of all numbers where \( -1 < x \leq 5 \). Therefore, set C is expressed as \( C = \{ x | -1 < x \leq 5 \} \).

Step 3: Find the Union of Sets B and C

The union of sets B and C, denoted as \( B \cup C \), includes all elements that are in either set B or set C or in both. Since B is the set of numbers where \( x < 4 \) and C is the set where \( -1 < x \leq 5 \), any number less than 4 or between -1 and 5 (including 5) is included. So, \( B \cup C = \{ x | -1 < x \leq 5 \} \).

Step 4: Find the Intersection of Sets B and C

The intersection of sets B and C, denoted as \( B \cap C \), includes all elements common to both sets. B includes numbers less than 4, and C includes numbers between -1 and 5. Therefore, the common values are those greater than -1 but less than 4. So, \( B \cap C = \{ x | -1 < x < 4 \} \).

Key Concepts

Union of Sets

The union of two sets is a fundamental concept in set theory, and it essentially means combining all the elements from each set without any duplication. When we talk about the union of sets, we're looking at everything that is included in either one of the sets, or both.
In this exercise, we're finding the union of sets B and C, which is written as \( B \cup C \). Set B consists of all elements where \( x < 4 \), whereas set C includes elements where \( -1 < x \leq 5 \). The union \( B \cup C \) will include any number that is:

• Less than 4, because those are present in set B.
• Between -1 and 5, as these are included in set C. Even though numbers greater than 4 but less than or equal to 5 exist in C, they are not in B, so not all are part of the intersection. But since we’re doing a union, they are included.

Thus, combining these observations, the union of B and C gives us the set \( \{ x | -1 < x \leq 5 \} \). It’s like taking all elements from both B and C without having to worry if they repeat—they just appear once in the union.

Intersection of Sets

An intersection shows the common elements between two sets. For sets B and C, finding the intersection involves identifying numbers that belong to both sets simultaneously.
For set B, our elements are such that \( x < 4 \). Meanwhile, for set C, the elements are those where \( -1 < x \leq 5 \). The intersection \( B \cap C \) will therefore be the numbers that satisfy both conditions at once:

• They need to be greater than -1, a condition from set C.
• They must also be less than 4, a condition brought by set B.

As a result, the intersection of these two sets is \( \{ x | -1 < x < 4 \} \). Only those values that fit both places make it into the intersection, linguistically and mathematically constraining our choice of elements.

Inequalities

Inequalities allow us to define sets with conditions where numbers are compared in terms of larger or smaller. They form a crucial basis when working with sets in set theory.
In our example, each set, A, B, and C, is defined through inequalities:

• Set A has an inequality \( x \geq -2 \), meaning it includes all numbers greater than or equal to -2.
• Set B uses the inequality \( x < 4 \), thus it includes numbers strictly less than 4.
• Set C represents numbers that follow \( -1 < x \leq 5 \), those that are more than -1 but no greater than 5.

Inequalities play a pivotal role when determining intersecting or union sets, as they help specify which numbers precisely meet the required conditions to be included in the resulting set. They’re like guardrails, helping us stay within the allowable framework of numbers when comparing different sets.