Problem 39 - Calculus: One and Several Variables

Question

Sketch the set on a number line. \((-\infty,-1) \cup(-2, \infty)\).

Step-by-Step Solution

Answer

The set \((-\infty,-1) \cup(-2, \infty)\) will comprise of all values on the number line less than -1 and greater than -2, excluding -1 and -2 themselves. There will be a gap on the number line between -1 and -2.

Step 1 Interpretation of the Symbols

The first step is to understand the symbols. Throughout, open parentheses '(' or ')' are used, which means the boundaries are excluded. Firstly, (-\infty, -1) is an interval from negative infinity to -1, but the -1 is not included in the set. Secondly, (-2, \infty) is an interval from -2 to infinity, not including -2 itself.

Step 2 Plot First Interval on a Number Line

Draw a number line and place a point where -1 is located, but as -1 is not included in the set, put an open circle on -1. Draw a line to the left from the open circle extending towards -\infty as all the values less than -1 are included.

Step 3 Plot Second Interval on a Number Line

Now deal with the second interval (-2, \infty). Put an open circle on -2, as -2 is not included in the set. Draw a line to the right from this open circle extending towards \infty as all the values greater than -2 are included.

Step 4 Final Composed Plot

Combine both steps 2 and 3 on the same number line. Since -1 and -2 both have open circles (excluded), there is a small gap between them. This gap represents that the values between -2 and -1 are not part of the set.

Key Concepts

Open Intervals

Open intervals are a type of interval that exclude the endpoints. In mathematics, when we denote an interval with parentheses, such as \((-\infty, -1)\) or \((-2, \infty)\), we are indicating that the boundary values themselves are not included in that set.

• The notation \((a, b)\) refers to all the numbers between \(a\) and \(b\), but not including \(a\) or \(b\).
• When intervals extend to infinity, such as \((-\infty, a)\) or \((b, \infty)\), they imply a limitless continuation in that direction.

When plotting open intervals on a number line, open circles are used at the boundaries to visually demonstrate that these points are excluded from the interval.

Infinity in Sets

Infinity is used in mathematics to describe a quantity that is larger than any finite number or continues indefinitely. In the context of sets and intervals, infinity (In mathematics, infinity is not a real number, but rather a concept used to illustrate that numbers or sequences extend without bound. When used in interval notation, such as \((-\infty, b)\) or \((a, \infty)\), it represents continuous values that do not have an endpoint.

• \(-\infty\) indicates a boundless set in the negative direction.
• \(\infty\) implies a set continuing indefinitely in the positive direction.

It is crucial to note that infinity itself is never included in a set, so parentheses, not brackets, are always used when infinity is involved.

Set Notation

Set notation is a standard method used to represent collections of objects, numbers, or intervals. It provides a clear and efficient way to specify which elements belong to a particular set.

• Interval notation, such as \((-\infty, -1) \cup (-2, \infty)\), captures the idea of combining sets or intervals, using a union symbol \(\cup\) to denote "or." It indicates all numbers included in either the first or the second interval.
• The symbols \(\infty\) and \(-\infty\) are used to describe unlimited bounds, which are characterized by open intervals.

Set notation simplifies visualizing and working with ranges on a number line, allowing for easier manipulation and comprehension of the relationships between numbers and sets.

Intervals on Number Line

Intervals on a number line represent specific sets of numbers. By marking intervals, we can visually understand which numbers are included or excluded from a set.

• Open circles indicate excluded endpoints, as in the open intervals \((-\infty, -1)\) and \((-2, \infty)\).
• Shading the region between these circles highlights the numbers encompassed by the intervals.
• When plotting \((-\infty, -1)\) and \((-2, \infty)\) on the same line, a gap appears between \(-2\) and \(-1\), signifying those numbers are not included.

Number lines are a valuable tool for translating algebraic notation into a visual form, thus enhancing understanding and clarity of mathematical concepts.