Problem 44 - 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) \(A \cap C = \{-1 < x \leq 5\}\); (b) \(A \cap B = \{-2 \leq x < 4\}\).

Step 1: Understand the Set A

Set \( A \) is defined as \( \{x \mid x \geq -2\} \). This means \( A \) includes all real numbers greater than or equal to \(-2\).

Step 2: Understand the Set B

Set \( B \) is defined as \( \{x \mid x < 4\} \). This means \( B \) includes all real numbers less than \( 4 \).

Step 3: Understand the Set C

Set \( C \) is defined as \( \{x \mid -1 < x \leq 5\} \). This means \( C \) includes all real numbers greater than \(-1\) and up to \( 5 \), including \( 5 \).

Step 4: Find the Intersection of A and C

The intersection \( A \cap C \) contains elements that are both in \( A \) and in \( C \). Since \( A = \{x \mid x \geq -2\} \) and \( C = \{x \mid -1 < x \leq 5\} \), \(-1 < x \leq 5\) is the narrower interval. Therefore, \( A \cap C = \{x \mid -1 < x \leq 5\} \).

Step 5: Find the Intersection of A and B

The intersection \( A \cap B \) contains elements that are both in \( A \) and in \( B \). Since \( A = \{x \mid x \geq -2\} \) and \( B = \{x \mid x < 4\} \), the overlapping interval will be \( -2 \leq x < 4 \). Thus, \( A \cap B = \{x \mid -2 \leq x < 4\} \).

Key Concepts

Intersection of Sets

In set theory, the intersection of sets refers to the elements that are common to both sets. It's a way to find a new set by focusing on what two or more sets have in common.

For example, if you have two sets, set \( A \) and set \( B \), the intersection is denoted as \( A \cap B \). This means you take all the elements that are present in both sets \( A \) and \( B \).

In our exercise, we see that the intersection is used to find common ground between sets involving inequalities. Since sets are defined by inequalities, the intersection looks for numbers that satisfy all given conditions. For \( A \cap C \), we find the numbers that satisfy both \( x \geq -2 \) and \( -1 < x \leq 5 \), which simplifies to \( -1 < x \leq 5 \). This is because this range is the "tighter" or more restrictive portion of both conditions.

Inequalities in Sets

Inequalities form the building blocks for defining certain types of sets. They provide boundaries that describe which numbers belong in the set.

When a set is defined by an inequality, it tells us the restrictions on the elements of the set. For example, the set \( A \) in our problem is described by the inequality \( x \geq -2 \). This means every number greater than or equal to \(-2\) is included in this set.

• \( A = \{x \mid x \geq -2\} \)
• \( B = \{x \mid x < 4\} \)
• \( C = \{x \mid -1 < x \leq 5\} \)

Understanding these inequalities and how they intersect helps us find intersections. Combining inequalities logically helps delineate what common numbers remain in intersected sets.

Real Number Line

The real number line is a visual concept that helps us understand how numbers are situated relative to each other. It's a straight line where every point on the line corresponds to a real number.

When dealing with set theory, the real number line helps us visualize the elements of a set. For instance, if a set includes numbers greater than or equal to \(-2\), on the line, any point to the right of \(-2\) would be included.

Understanding the real number line is crucial when considering intersections, as it allows you to see at a glance which parts of "greater than", "less than", or "between" conditions overlap and influence set intersections.

Set Notation

Set notation is a formal language used to define and express sets clearly and succinctly. It uses various symbols to show membership, intersections, unions, and other operations in sets.

For instance, the notation \( \{x \mid x \geq -2\} \) describes a set by stating that \( x \) must be greater than or equal to \(-2\). The vertical bar \( \mid \) in the set notation means "such that".

• \( \cap \) for intersections of sets
• \( \cup \) for union of sets
• Curly brackets \( \{ \} \) for listing elements or stating conditions

Using accurate set notation ensures that anyone reading the problem understands the precise conditions under which elements are part of the set, making complex problems more manageable by providing a clear structure.