x̄ - > Axiom of Replacement & Axiom of Subsets

 The Axiom of Replacement is a principle in set theory that allows us to form a new set by applying a function to elements of an existing set. While the Axiom of Replacement is typically discussed within the context of set theory and mathematical logic, I can provide you with an example in R code that demonstrates the concept.

In R, we can use the `lapply()` function to simulate the Axiom of Replacement. The `lapply()` function applies a function to each element of a list or vector and returns a new list or vector containing the results.

Here's an example that demonstrates the Axiom of Replacement using the `lapply()` function in R:

```R

# Define a set of numbers

numbers <- c(1, 2, 3, 4, 5)

# Define a function that squares a number

square <- function(x) {

  return(x^2)

}

# Apply the function to each element of the set

squared_numbers <- lapply(numbers, square)

# Print the squared numbers

print(squared_numbers)

```

In this example, we have a set of numbers (`numbers`) and a function (`square`) that squares each number. We use `lapply()` to apply the `square()` function to each element of the `numbers` set. The result is a new list (`squared_numbers`) that contains the squared values of the original numbers.

Please note that this example is a simplified illustration of the concept and does not directly correspond to the formal mathematical definition of the Axiom of Replacement.

The Axiom of Subsets, also known as the Axiom of Specification or the Axiom of Separation, is a principle in set theory that allows us to form a new set containing elements from an existing set that satisfy a certain condition. In R, we can simulate the Axiom of Subsets using conditional indexing or filtering techniques.

Here's an example in R code that demonstrates the Axiom of Subsets:

```R

# Define a set of numbers

numbers <- 1:10

# Define a condition to filter the set

condition <- numbers %% 2 == 0  # Select even numbers

# Create a subset using the condition

even_numbers <- numbers[condition]

# Print the subset of even numbers

print(even_numbers)

```

In this example, we have a set of numbers from 1 to 10 (`numbers`). We define a condition using the modulo operator (`%%`) to check if each number is even (`numbers %% 2 == 0`). We apply this condition as an index to the `numbers` set, creating a subset that only contains the even numbers. The result is a new vector (`even_numbers`) that contains the even elements from the original set.

You can modify the condition according to your requirements to create subsets based on different criteria. This way, you can simulate the Axiom of Subsets in R by selectively extracting elements from an existing set that satisfy a specific condition.