Finding Symmetric Elements In Sets With Operations

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Hey guys! Let's dive into the fascinating world of finding symmetric elements within sets that have specific operations defined on them. This might sound a bit complex, but we'll break it down step by step. We're going to explore how to determine the symmetric element of a given element 's' within a set 'M' when a certain operation (represented by '∘') is applied. We'll tackle several cases, each with different sets and operations. So, buckle up and let's get started!

a) M = R, x ∘ y = xy + x + y, s ∈ {-3, 2, √2}

In this first scenario, we're dealing with the set of real numbers (R), and the operation '∘' is defined as x ∘ y = xy + x + y. Our goal is to find the symmetric element for each 's' in the set {-3, 2, √2}. Remember, the symmetric element of 's', which we'll call 's'', is the element that, when operated with 's', results in the identity element. But first, we need to find that identity element!

Finding the Identity Element

The identity element 'e' is that special number that doesn't change anything when we use our operation. In other words, for any real number 'x', we need to find 'e' such that x ∘ e = x and e ∘ x = x. Let's use the definition of our operation:

  • x ∘ e = xe + x + e = x

Now, let's solve for 'e'. We can rearrange the equation:

  • xe + e = 0
  • e(x + 1) = 0

For this to hold true for any 'x', 'e' must be 0. So, the identity element for this operation is e = 0. This is a crucial step, because without knowing the identity element, we can't determine the symmetric elements. You see, the identity element acts as the reference point, the 'zero' or 'one' (depending on the operation) to which we need to get back when combining an element with its symmetric counterpart.

Determining Symmetric Elements

Now, let's find the symmetric element 's'' for each given 's'. The symmetric element 's'' should satisfy the equation: s ∘ s' = e = 0.

  • For s = -3:

    • -3 ∘ s' = -3s' - 3 + s' = 0
    • -2s' = 3
    • s' = -3/2

    So, the symmetric element of -3 is -3/2. We found this by plugging -3 into the equation and solving for the unknown 's''. This process essentially reverses the effect of -3 under the defined operation.

  • For s = 2:

    • 2 ∘ s' = 2s' + 2 + s' = 0
    • 3s' = -2
    • s' = -2/3

    Therefore, the symmetric element of 2 is -2/3. Notice how the symmetric element is not simply the negative of the original number. The operation dictates how the symmetric element is calculated.

  • For s = √2:

    • √2 ∘ s' = √2s' + √2 + s' = 0
    • s'(√2 + 1) = -√2
    • s' = -√2 / (√2 + 1)

    To rationalize the denominator, multiply the numerator and denominator by (√2 - 1):

    • s' = -√2(√2 - 1) / (2 - 1)
    • s' = -2 + √2

    Thus, the symmetric element of √2 is -2 + √2. This example shows that even with irrational numbers, the same principle applies. We just need to be careful with our algebraic manipulations.

Wrapping up Case a)

In this section, we successfully found the symmetric elements for -3, 2, and √2. The key takeaway here is the importance of first identifying the identity element. Then, we used the definition of the symmetric element (s ∘ s' = e) to set up equations and solve for the unknowns. Remember, the symmetric element undoes the effect of the original element under the specific operation.

b) M = Z, x ∘ y = x + y - 13, s ∈ {-1, 0, 3, 11}

Now, let's shift our focus to integers (Z) with a different operation: x ∘ y = x + y - 13. We need to find the symmetric elements for s ∈ {-1, 0, 3, 11}. Just like before, the first thing we need to do is determine the identity element.

Finding the Identity Element

We need to find 'e' such that x ∘ e = x for any integer 'x'. Let's use our operation:

  • x ∘ e = x + e - 13 = x

Now, solve for 'e':

  • e - 13 = 0
  • e = 13

So, the identity element for this operation is e = 13. Notice that the identity element is not necessarily 0 or 1; it depends entirely on the operation defined on the set.

Determining Symmetric Elements

Now, let's find the symmetric element 's'' for each given 's'. We need to solve the equation s ∘ s' = e = 13.

  • For s = -1:

    • -1 ∘ s' = -1 + s' - 13 = 13
    • s' = 27

    So, the symmetric element of -1 is 27.

  • For s = 0:

    • 0 ∘ s' = 0 + s' - 13 = 13
    • s' = 26

    Therefore, the symmetric element of 0 is 26.

  • For s = 3:

    • 3 ∘ s' = 3 + s' - 13 = 13
    • s' = 23

    Thus, the symmetric element of 3 is 23.

  • For s = 11:

    • 11 ∘ s' = 11 + s' - 13 = 13
    • s' = 15

    Therefore, the symmetric element of 11 is 15. It's crucial to check your answers by plugging them back into the original operation. This can help catch any calculation errors.

Wrapping up Case b)

In this case, we successfully found the symmetric elements within the set of integers under the given operation. Again, the process involved first identifying the identity element and then using the definition of the symmetric element to solve for the unknown. Notice how the symmetric elements are significantly different from the original elements, highlighting the importance of the specific operation in determining the symmetry.

c) M = C, x ∘ y = x + y + i, s ∈ {i, -i, 1+i}

Let's move on to complex numbers (C) with the operation: x ∘ y = x + y + i. We need to find the symmetric elements for s ∈ {i, -i, 1+i}. The procedure remains the same: find the identity element first.

Finding the Identity Element

We need to find 'e' in the set of complex numbers such that x ∘ e = x for any complex number 'x'.

  • x ∘ e = x + e + i = x

Solve for 'e':

  • e + i = 0
  • e = -i

So, the identity element is e = -i. Complex numbers have both real and imaginary parts, so the identity element can also be a complex number.

Determining Symmetric Elements

Now, find 's'' such that s ∘ s' = e = -i.

  • For s = i:

    • i ∘ s' = i + s' + i = -i
    • s' + 2i = -i
    • s' = -3i

    Thus, the symmetric element of i is -3i.

  • For s = -i:

    • -i ∘ s' = -i + s' + i = -i
    • s' = -i

    So, the symmetric element of -i is -i. In some cases, the symmetric element can be the element itself.

  • For s = 1 + i:

    • (1 + i) ∘ s' = (1 + i) + s' + i = -i
    • 1 + s' + 2i = -i
    • s' = -1 - 3i

    Therefore, the symmetric element of 1 + i is -1 - 3i. Complex number operations require careful attention to both the real and imaginary parts.

Wrapping up Case c)

We've now successfully navigated complex numbers and their symmetric elements. The key is to treat the imaginary unit 'i' with care and to correctly apply the operation's definition. Remember that the identity element and symmetric elements can be complex numbers themselves.

d) M = (-3, 3), x ∘ y = (9x + 9y) / (9 + xy)

Finally, let's tackle the interval (-3, 3) with the operation: x ∘ y = (9x + 9y) / (9 + xy). This operation looks a bit more involved, but the fundamental principle remains the same. We need to find symmetric elements, and to do that, we need the identity element first.

Finding the Identity Element

We need to find 'e' within the interval (-3, 3) such that x ∘ e = x for any 'x' in (-3, 3).

  • x ∘ e = (9x + 9e) / (9 + xe) = x

Let's solve for 'e'. Multiply both sides by (9 + xe):

  • 9x + 9e = x(9 + xe)
  • 9x + 9e = 9x + x^2e
  • 9e = x^2e
  • 9e - x^2e = 0
  • e(9 - x^2) = 0

For this to hold true for all 'x' in (-3, 3), 'e' must be 0 (since 9 - x^2 is not always zero in this interval). So, the identity element is e = 0.

Determining Symmetric Elements

Now, find 's'' such that s ∘ s' = e = 0.

  • s ∘ s' = (9s + 9s') / (9 + ss') = 0

For this fraction to be zero, the numerator must be zero:

  • 9s + 9s' = 0
  • 9s' = -9s
  • s' = -s

This is interesting! The symmetric element s' is simply the negative of s. This makes our work much easier. This is a great example of how the form of the operation can directly influence the relationship between an element and its symmetric counterpart.

  • For s: The symmetric element is -s. Since M = (-3, 3), and if s is in this interval, then -s is also guaranteed to be in this interval.

Wrapping up Case d)

In this final case, we discovered a particularly elegant result: the symmetric element is simply the negative of the original element. This was a direct consequence of the specific operation defined on the interval (-3, 3). This highlights the importance of paying close attention to the operation's form, as it can sometimes lead to simplified relationships.

Final Thoughts

Throughout these examples, we've explored how to find symmetric elements in different sets with various operations. The core steps are always the same: first, find the identity element; then, use the definition of the symmetric element to solve for the unknown. Remember that the identity element and the form of the operation are crucial in determining the symmetric elements. Keep practicing, guys, and you'll become pros at this in no time! Understanding these concepts is fundamental in abstract algebra and provides a solid foundation for more advanced mathematical topics. Keep up the great work!