Functions And Relations: Set A To A

Hey guys! Let's dive into some cool math problems involving sets, relations, and functions. Today, we're tackling a problem where we have a set A and a relation R defined on it. It sounds a bit complicated, but trust me, we'll break it down step by step. So, let's get started!

Understanding the Problem

We're given a relation R: A → A, where A = {-2, 1, 2, 3}. This means we're looking at how elements in set A relate to other elements within the same set A. The big question is: which of the following statements are true?

• The number of functions that can be created from set A to A is 64.
• The number of one-to-one correspondence functions (bijections) from A to A is 24.

Looks like we need to figure out how many functions and one-to-one correspondences we can make from set A to itself. Let's break down each part.

Functions from A to A

First, let's talk about functions. A function from A to A essentially assigns each element in A to another element in A. Think of it like a machine: you put in an element from A, and it spits out another element from A. The key thing about a function is that each input has exactly one output. No ambiguity allowed!

So, how many functions can we create? Well, let's consider each element in A. The first element, -2, can be mapped to any of the four elements in A (-2, 1, 2, or 3). That's four choices. Similarly, the element 1 can also be mapped to any of the four elements in A. Again, four choices. The same goes for the elements 2 and 3 – each has four possible mappings. Because each of these choices is independent, we multiply the number of possibilities together.

Therefore, the total number of functions from A to A is 4 * 4 * 4 * 4 = 4⁴ = 256. This is because each of the four elements in the domain A has four possible choices in the codomain A. So, the first statement, "The number of functions that can be created from set A to A is 64," is incorrect. The correct number of functions is actually 256. Understanding the fundamental principles of function mapping is essential to grasping this concept. When dealing with functions, remember that each element in the domain must have a unique image in the codomain, but multiple elements in the domain can map to the same element in the codomain. This principle is crucial in determining the total number of possible functions. In our case, since each of the four elements in set A can be mapped to any of the four elements in set A, we have 4 options for each element. Multiplying these options together (4 * 4 * 4 * 4) gives us the total number of functions, which is 256, not 64.

One-to-One Correspondence (Bijections) from A to A

Now, let's move on to one-to-one correspondences, also known as bijections. A bijection is a special type of function that is both injective (one-to-one) and surjective (onto). Let's break that down:

• Injective (One-to-One): This means that each element in A maps to a unique element in A. No two elements in A can map to the same element in A. In simpler terms, if a ≠ b, then f(a) ≠ f(b).
• Surjective (Onto): This means that every element in A is mapped to by at least one element in A. In other words, the range of the function is equal to the entire codomain (which is A in this case).

So, for a function to be a bijection from A to A, it needs to be both one-to-one and onto. Think of it as a perfect pairing - each element in the first set is paired with exactly one unique element in the second set, and there are no leftover elements in either set.

How many bijections can we create? Well, let's think about it like arranging the elements of A. For the first element, -2, we have 4 choices of where to map it. But once we've chosen a mapping for -2, we only have 3 choices left for the element 1, because it can't map to the same element that -2 maps to (that would violate the one-to-one rule!). Then, for the element 2, we only have 2 choices left, and finally, for the element 3, we only have 1 choice left. So, the total number of bijections is 4 * 3 * 2 * 1 = 24. This is also known as 4 factorial, written as 4!.

Therefore, the second statement, "The number of one-to-one correspondence functions (bijections) from A to A is 24," is correct! To fully grasp the concept of bijections, understanding both injectivity (one-to-one) and surjectivity (onto) is crucial. Injectivity ensures that each element in the domain maps to a unique element in the codomain, while surjectivity ensures that every element in the codomain is mapped to by at least one element in the domain. When a function satisfies both these conditions, it is a bijection. In simpler terms, a bijection is a perfect pairing between two sets, where each element in the first set is paired with exactly one unique element in the second set, and vice versa. Applying this to our problem, we determined that there are 24 possible ways to create such perfect pairings from set A to itself. This calculation is based on the factorial of the number of elements in the set (4! = 4 * 3 * 2 * 1 = 24), which represents the number of ways to arrange those elements in a one-to-one and onto manner.

Conclusion

Alright, guys! We've tackled this problem and found that:

• The number of functions from A to A is 256 (not 64).
• The number of one-to-one correspondence functions (bijections) from A to A is indeed 24.

So, the second statement is the only correct one. I hope this explanation helps you understand the concepts of functions and bijections a bit better. Keep practicing, and you'll get the hang of it in no time! This problem highlights the importance of understanding the definitions and properties of functions, particularly the distinctions between general functions and bijections. By carefully considering the constraints imposed by these definitions, we can accurately determine the number of possible mappings between sets. In this case, the key was to recognize that a bijection requires a perfect pairing of elements, which leads to the factorial calculation.