Question

Let \(A=[n: n \in \mathbb{N}\) and \(n=2 k+1\) for some \(k \in \mathbb{N}\\}, B=\mid n: n \in \mathbb{N}\) and \(n=\) \(4 k+1\) for some \(k \in \mathbb{N}\\},\) and \(C=\\{m \in \mathbb{N}: m=2 k-1\) and \(k \in \mathbb{N}\) and \(k \geq 11\). Prove the following: (a) \(35 \in A\) (b) \(35 \in C\) (c) \(35 \notin B\) (d) \(A=C\) (c) \(B \subseteq A\) (f) \(B \subseteq C\) (g) \(B \subset A\) (h) \(B \subset C\)

Short answer

35 is in both sets A and C, but not in B. Set A equals set C, and B is a subset of both A and C, but strictly.

Step-by-step solution

Understanding Set A

The set \( A \) includes natural numbers \( n \) such that \( n = 2k+1 \) for some natural number \( k \). Essentially, \( A \) comprises all odd natural numbers.

Understanding Set B

The set \( B \) includes natural numbers \( n \) such that \( n = 4k+1 \). These are numbers that are 1 more than a multiple of 4.

Understanding Set C

The set \( C \) includes natural numbers \( m \) such that \( m = 2k-1 \) for \( k \geq 11 \). These are odd numbers starting from 21 (when \( k = 11 \)) and continuing upwards.

Prove that 35 is in Set A

Let \( n = 35 \). We need to find \( k \) such that \( 35 = 2k + 1 \). Solving for \( k \), we get: \( 35 - 1 = 2k \), so \( 34 = 2k \). Hence, \( k = 17 \). Thus, 35 is in set \( A \).

Prove that 35 is in Set C

Let \( m = 35 \). We need to find \( k \) such that \( 35 = 2k - 1 \). Solving for \( k \), we get: \( 35 + 1 = 2k \), so \( 36 = 2k \). Hence, \( k = 18 \), which satisfies \( k \geq 11 \). Thus, 35 is in set \( C \).

Prove that 35 is not in Set B

Let \( n = 35 \). We need to find \( k \) such that \( 35 = 4k + 1 \). Solving for \( k \), we get: \( 35 - 1 = 4k \), so \( 34 = 4k \). There is no integer \( k \) such that \( 4k = 34 \) (since \( 34/4 = 8.5 \)), meaning 35 is not in set \( B \).

Prove A equals C

Since both sets \( A \) and \( C \) start from number 21 onwards and include all odd numbers, they are equivalent. Thus, \( A = C \).

Prove B is a subset of A

Every element in \( B \) can be expressed as \( 4k+1 \), making it an odd number since \( 4 \times k \) is always even and adding 1 results in an odd number. Hence, every element of \( B \) also lies in \( A \), proving \( B \subseteq A \).

Prove B is a subset of C

The minimum in set \( B \) that fits \/ (\ is 1 more than a multiple of 4, which starts at 5 and increases as \( 4k+1 \) for \( k \geq 11 \). Since all such numbers are odd and greater than or equal to 21, they match the definition of \( C \), hence \( B \subseteq C \).

Prove B is a strict subset of A

Set \( B \) has only numbers of the form \( 4k+1 \), whereas \( A \) includes all odd numbers. Since \( B \) is a subset of the odd numbers but not all odd numbers are in \( B \), \( B \) is strictly a subset of \( A \), therefore \( B \subset A \).

Prove B is a strict subset of C

Similarly, \( B \) is made up of numbers \( 4k+1 \), while \( C \) comprises odd numbers calculated from odd values \( \geq 21 \). Since not all such numbers fit the pattern of \( 4k+1 \), \( B \) is strictly a subset of \( C \), thus \( B \subset C \).

Key concepts

Odd Numbers

Odd numbers are integers that are not divisible by 2. These numbers can be expressed in the form \( 2k + 1 \), where \( k \) is an integer. This expression is essential in identifying odd numbers because it signifies that after multiplying \( k \) by 2 (which yields an even result), adding 1 makes the outcome odd.

Odd numbers have a few defining characteristics:

• Adjacent odd numbers have a consistent difference of 2. For instance, 3 and 5 or 9 and 11.
• When an odd number is squared, the result is always odd. Similarly, multiplying two odd numbers results in an odd number.

In terms of set theory, the set of odd numbers is often represented in expressions such as \( 2k + 1 \) to form a basis for identifying numbers belonging to a specific pattern or rule, such as those in exercise sets like \( A \) and \( C \).

Thus, understanding odd numbers and how to represent them algebraically helps in various mathematical proofs and problems, such as those dealt with in this exercise.

Subsets

In set theory, a subset is essentially a set whose elements are entirely contained within another set, referred to as the superset. If set \( B \) is a subset of set \( A \), then every element in \( B \) is also in \( A \). This is denoted by \( B \subseteq A \).

To determine if one set is a proper subset of another, each element of the initial set must also be in the second set while ensuring there are elements in the second set not found in the first. In our exercise, \( B \) is shown to be a proper subset of \( A \) as \( B \subset A \) because \( B \) consists of numbers expressed as \( 4k + 1 \), making them odd, which automatically places them within the broader category of all odd numbers in \( A \).

Furthermore, there are higher implications of subsets when proving relationships between sets in mathematical theories. Set \( B \) can be seen as a proper subset of sets \( C \) as well, reinforcing the importance of set member criteria derived from expressions such as \( 4k+1 \), for structured problem solving and proofs.

Mathematical Proofs

Mathematical proof is a logical argument that verifies the truth of a mathematical statement beyond doubt. Proofs play a crucial role in mathematics, ensuring that a statement or theorem is universally valid.

There are several types of proofs, including:

• Direct Proofs: These involve straightforward steps that lead from the premises of the theorem to the conclusion. Each step is logically derived, as seen when solving for instance, proof that 35 belongs to set \( A \).
• Indirect Proofs: Also known as proof by contradiction, this method involves assuming the negation of the statement and reaching a contradiction, proving that the original assumption must be true.
• Proof by Induction: Often used in proofs concerning integers, it shows that if a proposition holds for one integer, it holds for the next, thus concluding it holds for all integers.

In our exercise, we used direct proofs to establish which sets the number 35 belongs to or does not belong to. Each proof step demonstrated either inclusion or exclusion through calculations based on the given form of numbers in sets. Each step serves to logically cement the relationships and subsets defined in the problem statement.