Question

Prove that \(\\{12 a+4 b: a, b \in \mathbb{Z}\\}=\\{4 c: c \in \mathbb{Z}\\}\).

Short answer

Based on the explorations and substantiations made above, it can be conclusively stated that \(\{12 a+4 b: a, b \in \mathbb{Z}\}=\{4 c: c \in \mathbb{Z}\}\). This is because we can represent any value generated by the expression '12a + 4b' as '4c', where 'a', 'b', and 'c' are integers. So, the sets generated by these two expressions are equal and identical.

Step-by-step solution

Formulating an assumption

Assume that \(2a + b = c\). Here 'c' is just another integer. This implies that we can represent \(12a + 4b\) (or equivalently, \(4(3a) + 4b\)) as \(4c\). As 'c' could be any integer, \(4c\) is capable of taking any value that \(12a + 4b\) can.

Initial condition test

To confirm the claim, let's show that the equation will hold for an initial condition. For instance, we can plug in a = 0 and b = 0 into '12a + 4b', which will give us 0. Simultaneously, if we plug in c = 0 into '4c', it will also give us 0. This shows that for at least one case, our assumption was correct.

Formulating the reverse assumption

Now, let's assume that there's an integer 'c' such that it cannot be represented as \(12a + 4b\). This would imply there's an integer 'c' such that \(4c\) cannot be written as \(4(3a+b)\), or more simplified, 'c' cannot be written as \(3a+b\). Considering 'a' and 'b' could be any integers, this is clearly not true.

Conclusion

Therefore, based on our manipulation and assumptions, we can conclusively say that \(\{12 a+4 b: a, b \in \mathbb{Z}\}=\{4 c: c \in \mathbb{Z}\}\). If we choose any integer value for 'a' and 'b', and apply them to the expression '12a + 4b', then there certainly is an integer 'c' that when applied to the expression '4c' will generate the same result. Hence, the two sets are indeed identical.

Key concepts

Proof by Assumption

Proof by assumption is a technique often used in mathematics to establish the validity of a proposition by assuming its truth and working through logical steps to show that no contradictions arise from the assumption. Indeed, this method sets the foundation for proving statements about integers, sets, and various other mathematical constructs.

In our exercise, we start by assuming that an integer 'c' can be represented as a combination of other integers 'a' and 'b'. We then explore this assumption from two angles: one where we hypothesize 'c' can take the form '12a + 4b', and the other where it cannot. By examining the potential outcomes, we see that if our assumptions hold true in at least one instance and no contradictions are found overall, we can confidently state the original proposition is correct.

Integer Properties

Integer properties play an essential role in algebraic manipulations, especially when dealing with equations and proofs involving whole numbers. Some key properties include closure (the sum or product of any two integers is an integer), commutativity (the order in which you add or multiply integers does not change the result), and distributivity (you can distribute multiplication over addition or subtraction).

In our exercise, when we examine the expressions '12a + 4b' and '4c', we leverage these integer properties. We observe that multiplying or adding integers results in an integer, a factor that is critical in the simplification and comparison of the two sets in question. Additionally, it is these properties that assure us the reverse assumption step, where we consider if an integer 'c' can exist outside the formulation of '12a + 4b', leads to a contradiction because the set of all integers is closed under addition and multiplication.

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In mathematics, sets are typically denoted using curly braces and can include anything from numbers to more abstract elements.

Our exercise revolves around the equality of two sets of integers: the set generated by the expression '12a + 4b' for all integers 'a' and 'b', and the set generated by '4c' for all integers 'c'. To prove that these two sets are identical, we work through our assumptions and show that for every element in one set, there is a corresponding and unique element in the other set, and vice versa. This equality of sets is foundational in set theory, and by systematically verifying our assumption, we establish that the two given sets indeed contain the same elements - a result that reflects the deep interconnections between set theory and other areas of mathematics.