Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Exponents

College Algebra

0. Review of Algebra

Exponents

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1:40 minutes

Problem 67

Textbook Question

Insert ⊆ or s in each blank to make the resulting statement true. ∅ ____ {1, 4, 6, 8}

Verified step by step guidance

Identify the symbols: The symbol '⊆' stands for 'is a subset of', and the symbol 's' is not a standard set theory symbol.

Understand the empty set: The empty set, denoted by '∅', contains no elements.

Consider the set {1, 4, 6, 8}: This set contains the elements 1, 4, 6, and 8.

Apply the subset definition: A set A is a subset of a set B if every element of A is also an element of B.

Determine the relationship: Since the empty set has no elements, it does not contain any elements that are not in {1, 4, 6, 8}, thus '∅' is a subset of {1, 4, 6, 8}.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Set Notation

Set notation is a mathematical language used to describe collections of objects, known as sets. The symbols used, such as '∅' for the empty set and '{1, 4, 6, 8}' for a set containing specific elements, help convey relationships between different sets. Understanding how to read and interpret these symbols is crucial for solving problems involving sets.

Subset

A subset is a set where all elements are contained within another set. The notation 'A ⊆ B' indicates that set A is a subset of set B, meaning every element of A is also an element of B. The empty set '∅' is a unique subset of every set, as it contains no elements and thus satisfies the subset condition for any set.

Empty Set

The empty set, denoted as '∅', is a fundamental concept in set theory representing a set with no elements. It plays a critical role in various mathematical contexts, including the definition of subsets and the concept of union and intersection. Recognizing that the empty set is a subset of any set is essential for understanding relationships between sets.