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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2:15 minutes

Problem 68

Textbook Question

Insert ⊆ or s in each blank to make the resulting statement true. ∅ ____ ∅

Verified step by step guidance

Identify the symbols: ∅ represents the empty set, which contains no elements.

Understand the symbols ⊆ and ⊂: The symbol ⊆ means 'is a subset of' and can include the scenario where two sets are equal. The symbol ⊂ means 'is a proper subset of' and is used when the subset is strictly contained within the other set and they are not equal.

Apply the definitions to the empty set: The empty set is a subset of every set, including itself. This is because there are no elements in the empty set to contradict the definition of being a subset.

Determine the correct symbol: Since ∅ is a subset of itself and they are equal, the symbol that includes equality should be used.

Choose the appropriate symbol: The correct symbol to insert is ⊆, indicating that the empty set is a subset of itself, including the possibility of equality.

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Key Concepts

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

Subset

A set A is considered a subset of a set B if every element of A is also an element of B. This relationship is denoted as A ⊆ B. The empty set, denoted as ∅, is a subset of every set, including itself, because it contains no elements that could contradict this definition.

Empty Set

The empty set, represented as ∅, is a unique set that contains no elements. It is a fundamental concept in set theory, serving as the identity element for the operation of union and a subset of every set. Understanding the properties of the empty set is crucial for grasping more complex set relationships.

Set Notation

Set notation is a mathematical language used to describe sets and their relationships. Symbols like ⊆ (subset) and ∅ (empty set) are part of this notation. Familiarity with these symbols allows for clear communication of set relationships and is essential for solving problems involving sets.