Problem 15 - Calculus Single Variable

Question

Write the set using interval notation. Use the symbol \(\cup\) where appropriate. \(\\{t: t>1\\}\)

Step-by-Step Solution

Answer

\((1, \, \infty)\)

Step 1: Identify the type of interval

The problem asks for the set of all numbers greater than 1. This implies an interval that does not include 1, and extends infinitely in the positive direction.

Step 2: Determine the interval bounds and type

Since the inequality is strictly greater than 1 (i.e., not equal to 1), the interval starts just over 1 and continues to infinity. The appropriate interval notation excludes 1, so we use a parenthesis, not a bracket.

Step 3: Write the interval in interval notation

The numbers we want go from just over 1 (not including 1) to infinity. In interval notation, we write this as \((1, \, \infty)\).

Key Concepts

Inequalities

Inequalities are like mathematical expressions that show the relationship between quantities. They tell you if one number is less than, greater than, equal to, or not equal to another number. For example, the inequality \(t > 1\) means that \(t\) is any value greater than 1.

• The symbol \(>\) means "greater than."
• The symbol \(<\) means "less than."
• The symbol \(\geq\) means "greater than or equal to."
• The symbol \(\leq\) means "less than or equal to."

Inequalities are essential because they describe ranges of numbers rather than specific values. This can be very useful when dealing with real-world problems. For example, if you have a budget of $1,000, you might want to spend less than or equal to that amount on a new computer. This would be represented by the inequality \(x \leq 1000\). Understanding inequalities allows you to set conditions and make informed decisions.

Infinite Intervals

Infinite intervals are intervals that extend indefinitely in either the positive or negative direction of the number line. In mathematics, these intervals help express ranges that don't have an end point. For example, when you say \(t > 1\), you are indicating that \(t\) can be any value greater than 1, all the way up to infinity.

• An interval extending to positive infinity is written as \((a, \infty)\), meaning it starts from \(a\) and goes beyond.
• An interval stretching to negative infinity is written as \((-\infty, b)\), which means it ends at \(b\) from the negative direction.

When you use intervals that extend to infinity, always remember to use parenthesis \((\) instead of brackets \([\), because infinity is not an actual number, and thus cannot be included in the interval. This is what we see in our original exercise where the set notation \(\{t: t>1\}\) is expressed using the interval notation \((1, \infty)\). This denotes all numbers greater than 1 without end.

Set Notation

Set notation is a simple way to describe a collection of elements, usually numbers, that satisfy a certain condition. This is often used in mathematics to define particular groups of numbers. The notation \(\{t: t>1\}\) is read as "the set of all \(t\) such that \(t\) is greater than 1."

• Braces \(\{\}\) are used to define the set.
• A colon \(:\) or vertical bar \(\mid\) separates the variable from its condition.

Set notation offers an efficient and concise way to represent inequalities or conditions without listing all possible elements. In our case, the set \(\{t: t>1\}\) represents an infinite group of numbers greater than 1, starting just beyond 1 and including everything larger. This form of notation is especially useful in mathematical proofs and computations when general descriptions of numbers are more practical than individual listings.