Problem 55
Question
Let \(a_{n}\) denote the number of subsets of the set \(S=\\{1,2, \ldots, n\\}\) that contain no consecutive integers, where \(n \geq 0 .\) When \(n=0, S=\emptyset\). Compute each. $$a_{0}$$
Step-by-Step Solution
Verified Answer
The given set is the empty set with n = 0, and it only has one subset, which is the empty set itself. This subset doesn't contain any consecutive integers. Therefore, \(a_{0} = 1\).
1Step 1: When n = 0, the set S is the empty set, written as S = ∅. #Step 2: Find subsets of the empty set#
Since S is an empty set, it has only one subset, which is the empty set itself. So the only subset of S is {∅}.
#Step 3: Check if subsets contain consecutive integers#
2Step 2: Now we need to check if the subsets contain any consecutive integers. Since there is only one subset (the empty set), and it contains no elements, it surely doesn't contain any consecutive integers. #Step 4: Compute the value of \(a_{0}\)#
Since the empty set is the only subset of the empty set with no consecutive integers, the value of \(a_{0}\) is 1.
Thus, \(a_{0} = 1\).
Key Concepts
SubsetsEmpty SetConsecutive Integers
Subsets
A subset is any possible selection of elements from a given set. For instance, if you have a set containing the elements \( \{a, b\} \), the possible subsets are \( \{\}, \{a\}, \{b\}, \text{and} \{a, b\} \). Subsets can range from the empty subset, containing no elements, to the full set itself, containing all elements.
When you are asked to find subsets, you consider all different ways you can choose each element in the original set to either be included or not included in any given subset.
Since the number of subsets is influenced by the number of elements in the set, it helps to remember that a set with \( n \) elements has \( 2^n \) subsets. This is because each element can be either included or excluded from any subset, giving two options per element. This calculation covers all possible combinations, from the empty set to the full set.
When you are asked to find subsets, you consider all different ways you can choose each element in the original set to either be included or not included in any given subset.
Since the number of subsets is influenced by the number of elements in the set, it helps to remember that a set with \( n \) elements has \( 2^n \) subsets. This is because each element can be either included or excluded from any subset, giving two options per element. This calculation covers all possible combinations, from the empty set to the full set.
Empty Set
The empty set, denoted by \( \emptyset \), is a unique concept in set theory. It is the set that contains no elements whatsoever. This may seem a bit abstract, but it is a fundamental part of set theory and mathematics as a whole.
Essentially, the empty set is like a basket that holds nothing. Despite holding no elements, it is still considered a valid set. This makes it particularly intriguing, as it is the only set with a size or cardinality of zero.
Its unique properties make it the smallest set since no subset can have fewer elements than zero. Interestingly, the empty set is also a subset of every possible set. This is because there are no elements in \( \emptyset \) to conflict with any elements in any other set.
Essentially, the empty set is like a basket that holds nothing. Despite holding no elements, it is still considered a valid set. This makes it particularly intriguing, as it is the only set with a size or cardinality of zero.
Its unique properties make it the smallest set since no subset can have fewer elements than zero. Interestingly, the empty set is also a subset of every possible set. This is because there are no elements in \( \emptyset \) to conflict with any elements in any other set.
Consecutive Integers
Consecutive integers are numbers that follow each other in sequence without any gaps between them. For example, in the sequence of numbers, 3, 4, and 5 are consecutive integers. Similarly, 10 and 11 are consecutive, as there is no integer missing between them.
In problems involving sets of numbers, having no consecutive integers means ensuring that no two numbers in a subset are directly next to each other in value. If you take a set \( \{1, 3, 4\} \), you see that 3 and 4 are consecutive, and hence, should not exist together if the rule is to have no consecutive integers in the subset.
This concept is more than just a counting exercise; it's about understanding how to selectively group numbers without certain pairs. Knowing whether specific sequences are allowed can influence the way you generate number sets or prove combinatorial results.
In problems involving sets of numbers, having no consecutive integers means ensuring that no two numbers in a subset are directly next to each other in value. If you take a set \( \{1, 3, 4\} \), you see that 3 and 4 are consecutive, and hence, should not exist together if the rule is to have no consecutive integers in the subset.
This concept is more than just a counting exercise; it's about understanding how to selectively group numbers without certain pairs. Knowing whether specific sequences are allowed can influence the way you generate number sets or prove combinatorial results.
Other exercises in this chapter
Problem 54
Let \(a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{N},\) where \(n \geq 2\) . Prove that \(\operatorname{gcd}\left\\{a_{1}, a_{2}, \dots, a_{n}\right\\}=\operatornam
View solution Problem 54
[These exercises indicate that \(u_{n}=L_{n},\) the \(n\) th Lucas number. Accordingly, \(\left.u_{n}=\alpha^{n}+\beta^{n} \text { is the Binet form of } L_{n}
View solution Problem 56
Let \(f(n)=2 f(n / 2)+c n^{2},\) where \(f(1)=d\) and \(n\) is a power of 2 Solve the recurrence relation.
View solution Problem 56
Let \(a_{n}\) denote the number of subsets of the set \(S=\\{1,2, \ldots, n\\}\) that contain no consecutive integers, where \(n \geq 0 .\) When \(n=0, S=\empty
View solution