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Chapter 1: Problem 13

Let \(A\) and \(B\) be sets with the property that there are exactly 144 sets which are subsets of at least one of \(A\) or \(B\). How many elements does the union of \(A\) and \(B\) have?

Short Answer

Expert verified
144 sets imply 7 elements, but direct fit suggests re-evaluation.

Step by step solution

01

Understand the Problem

We need to find the number of elements in the union of sets A and B given that there are exactly 144 subsets that are subsets of either A or B, or both.
02

Recall the Formula for Number of Subsets

The number of subsets of a set with n elements is given by the formula \(2^n\), where n is the number of elements in the set.
03

Represent Total Subsets

Let \(|A \cup B| = n\). The total number of subsets of the union \(A \cup B\) with n elements is \(2^n\). We know this total is 144, so we have the equation: \(2^n = 144\).
04

Solve for n

To find n, consider that \(2^7 = 128\) and \(2^8 = 256\). Hence, there is no integer n such that \(2^n = 144\). Reconsideration or alternative approach might be needed.
05

Reinterpret the Problem

Given the problem and its set notation context, note potential reinterpretations in terms of proper subset counts or different base set handling.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subsets
A subset is a set that contains some or all elements of another set. For example, if we have a set A = {1, 2, 3}, its subsets include {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}.
Remember that the empty set is always a subset of any set.
When working with subsets, the key formula to remember is that a set with n elements has exactly \(2^n\) subsets. This is because each element has two possibilities: either it is included in a subset or it is not.
For example, if A has 3 elements, then it has \(2^3 = 8\) subsets. Similarly, if A has n elements, it will have \(2^n\) subsets.
In our problem, this concept helps us determine the total number of possible subsets of the union of sets A and B.
Union of Sets
The union of two sets A and B, denoted by \(A \cup B\), is a new set that contains all the elements that are in A, in B, or in both.
For example, if A = {1, 2, 3} and B = {3, 4}, then \(A \cup B\) = {1, 2, 3, 4}.
When dealing with union in set theory, an important aspect is counting the unique elements. If we know the number of elements in each set and how they overlap, we can utilize the principle of inclusion-exclusion. However, for this problem, we need to recognize that the total number of subsets of the union must fit the given number.
Specifically, we are given that there are 144 subsets in total, which means that if n is the number of elements in \(A \cup B\), then \(2^n = 144\).
Unfortunately, since 144 is not a power of 2, reconsidering the interpretation or assumptions made about the sets might be necessary.
Combinatorics
Combinatorics is a field of mathematics focused on counting, arrangement, and combination of objects.
In our problem, we use combinatorial principles to understand the number of subsets formed by the elements of sets A and B.
We need to carefully consider how we counted subsets and whether additional conditions or interpretations should be addressed.
Through combinatoric analysis, we revisit our steps: We identified the formula for subsets, \(2^n\). We noted there appeared no suitable integer for \(2^n = 144\). We then recall reevaluation might reveal overlooked aspects or better suited interpretations for unique sets properties.

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