Question

If \(\mathrm{A}=\left\\{\mathrm{n}^{3}+(\mathrm{n}+1)^{3}+(\mathrm{n}+2)^{3} ; \mathrm{n} \in \mathrm{N}\right\\}\) and \(\mathrm{B}=\\{9 \mathrm{n}, \mathrm{n} \in \mathrm{N}\\}\) then (a) \(A \subset B\) (b) \(\mathrm{B} \subset \mathrm{A}\) (c) \(\mathrm{A}=\mathrm{B}\) (d) \(\mathrm{A}^{\prime}=\mathrm{B}\)

Short answer

The answer is (a) \(A \subset B\).

Step-by-step solution

Write down the given sets

Set A: \(A = \{n^3 + (n + 1)^3 + (n + 2)^3; n \in N\}\) Set B: \(B = \{9n; n \in N\}\)

Analyze A's elements

Consider the elements of set A: \(n^3 + (n + 1)^3 + (n + 2)^3\), expand the expression for better understanding. \(n^3 + (n^3 + 3n^2 + 3n + 1) + (n^3 + 6n^2 + 12n + 8) \\ = 3n^3 + 9n^2 + 15n + 9\) So the elements of set A are in the form of \(3n^3 + 9n^2 + 15n + 9\).

Analyze B's elements

Now consider the elements of set B: \(9n\), this implies that the elements are multiples of 9.

Compare A and B

Now we need to determine if the elements in A are multiples of 9, which is a condition for B's elements. For elements of the set A to be multiples of 9, the expression \(3n^3 + 9n^2 + 15n + 9\) must be divisible by 9. The expression can be factored as: \(9(1 + n)(n^2 + 2)\) Since \(9(1 + n)(n^2 + 2)\) is always divisible by 9, that means any element in set A is also an element of set B.

Conclusion

Since every element in A is also an element of B, we can say that \(A \subset B\). Therefore, the answer is (a) \(A \subset B\).

Key concepts

Set Notation

In mathematics, set notation is a standardized method of describing collections of objects, known as 'elements.' When we denote a set like \(A\), we're referring to a specific group of elements that share a common property or rule for inclusion. For example, in set \(A\) from the exercise:
\[A = \{n^3 + (n + 1)^3 + (n + 2)^3; n \in \mathbb{N}\}\]
elements of \(A\) are determined by a formula involving natural numbers (\(\mathbb{N}\)). Each element is the sum of three consecutive natural numbers each raised to the third power. Set \(B\) is simpler:
\[B = \{9n; n \in \mathbb{N}\}\]
where each element is a multiple of 9. Using set notation efficiently communicates the structure and elements of each set, making it easier to analyze and compare them.

Cubic Expansion

Cubic expansion concerns the process of expanding algebraic expressions raised to the third power. In our example, we expand the compact form \( (n + 1)^3 \) and \( (n + 2)^3 \) to understand how elements in set \(A\) are constructed. By expanding these expressions, we can see the pattern or structure of the numbers forming the set. When expanding \( (n + 1)^3 \) and \( (n + 2)^3 \) stepwise, we rely on the binomial theorem and identity:\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
This procedure helps identify whether set \(\mathbb{N}\) includes certain characteristics like divisibility by a specific number—in this case, 9.

Divisibility

The concept of divisibility is central to many aspects of set theory and number theory. A number \( p \) is divisible by another number \( q \) if you can divide \( p \) by \( q \) with no remainder. In set \(B\), each element is of the form \(9n\), indicating that each element is divisible by 9. The divisibility rule for 9 is particularly useful in our problem: if the sum of the digits of a number is divisible by 9, then the number itself is also divisible by 9. By factoring out 9 from the expanded form of set \(A\)'s elements, we demonstrated that these elements too are divisible by 9, thus observing a critical relationship between sets \(A\) and \(B\).

Subset Relationship

The subset relationship is a fundamental concept in set theory, where one set is entirely contained within another set. We denote this relationship with the symbol \( \subseteq \). If every element of set \(A\) is also an element of set \(B\), then \(A\) is a subset of \(B\), expressed as \(A \subseteq B\). After expanding the cubic expressions in set \(A\) and verifying their divisibility by 9, we concluded that all elements of \(A\) are indeed elements of set \(B\). Thus, we can definitively say that \(A \subseteq B\), which is the correct answer to the original problem. Understanding subset relationships is crucial when dealing with more complex set operations and comparisons.