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Let's delve into the task step-by-step to understand how the data and labels are determined. Our data set contains two binary features, [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex], and the class label [tex]\(y\)[/tex] is determined by a condition similar to the XOR binary operator, except we're using [tex]\(-1\)[/tex] instead of 0.
Step 1: Understanding Binary Features
We need a description of how the features are aligned and what the possible combinations of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] could look like with a binary system of [tex]\(\{-1, 1\}\)[/tex]. The possible combinations for [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] would be:
- [tex]\(x_1 = 1\)[/tex], [tex]\(x_2 = 1\)[/tex]
- [tex]\(x_1 = -1\)[/tex], [tex]\(x_2 = 1\)[/tex]
- [tex]\(x_1 = 1\)[/tex], [tex]\(x_2 = -1\)[/tex]
- [tex]\(x_1 = -1\)[/tex], [tex]\(x_2 = -1\)[/tex]
Step 2: Determining the Label [tex]\(y\)[/tex]
The label [tex]\(y\)[/tex] is determined by the XOR operation adjusted for -1 and 1:
- For the classic XOR operation, (0 XOR 0=0, 1 XOR 0=1, 0 XOR 1=1, 1 XOR 1=0), we adjust the values to [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
The class label [tex]\(y\)[/tex] is determined by the product of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] following the rule:
- [tex]\(y = 1\)[/tex] if [tex]\(x_1 \times x_2\)[/tex] is positive (either both are 1 or both are -1)
- [tex]\(y = -1\)[/tex] if [tex]\(x_1 \times x_2\)[/tex] is negative (one is 1 and the other is -1)
Step 3: Applying the XOR Conditional Rule
Now, according to the XOR rules:
- When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( 1 \times 1 ) = 1\)[/tex].
- When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( -1 \times 1 ) = 1\)[/tex].
- When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( 1 \times -1 ) = 1\)[/tex].
- When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( -1 \times -1 ) = -1\)[/tex].
Step 4: Constructing the Data Set
Given the explanation above:
1. When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y = 1\)[/tex]
- [1, 1, 1]
2. When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y = 1\)[/tex]
- [-1, 1, 1]
3. When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y = 1\)[/tex]
- [1, -1, 1]
4. When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y = -1\)[/tex]
- [-1, -1, -1]
As a result, the data set with 2 binary features, [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex], and their corresponding class labels [tex]\(y\)[/tex] can be represented as follows:
[tex]\[ \begin{align*} \begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \\ -1 & -1 & -1 \\ \end{bmatrix} \][/tex]
This provides the detailed, step-by-step breakdown of how we arrive at the final data set:
[tex]\[ \begin{bmatrix} [1, 1, 1], \\ [-1, 1, 1], \\ [1, -1, 1], \\ [-1, -1, -1] \end{bmatrix} \][/tex]
Step 1: Understanding Binary Features
We need a description of how the features are aligned and what the possible combinations of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] could look like with a binary system of [tex]\(\{-1, 1\}\)[/tex]. The possible combinations for [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] would be:
- [tex]\(x_1 = 1\)[/tex], [tex]\(x_2 = 1\)[/tex]
- [tex]\(x_1 = -1\)[/tex], [tex]\(x_2 = 1\)[/tex]
- [tex]\(x_1 = 1\)[/tex], [tex]\(x_2 = -1\)[/tex]
- [tex]\(x_1 = -1\)[/tex], [tex]\(x_2 = -1\)[/tex]
Step 2: Determining the Label [tex]\(y\)[/tex]
The label [tex]\(y\)[/tex] is determined by the XOR operation adjusted for -1 and 1:
- For the classic XOR operation, (0 XOR 0=0, 1 XOR 0=1, 0 XOR 1=1, 1 XOR 1=0), we adjust the values to [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
The class label [tex]\(y\)[/tex] is determined by the product of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] following the rule:
- [tex]\(y = 1\)[/tex] if [tex]\(x_1 \times x_2\)[/tex] is positive (either both are 1 or both are -1)
- [tex]\(y = -1\)[/tex] if [tex]\(x_1 \times x_2\)[/tex] is negative (one is 1 and the other is -1)
Step 3: Applying the XOR Conditional Rule
Now, according to the XOR rules:
- When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( 1 \times 1 ) = 1\)[/tex].
- When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( -1 \times 1 ) = 1\)[/tex].
- When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( 1 \times -1 ) = 1\)[/tex].
- When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y\)[/tex] should be [tex]\(( -1 \times -1 ) = -1\)[/tex].
Step 4: Constructing the Data Set
Given the explanation above:
1. When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y = 1\)[/tex]
- [1, 1, 1]
2. When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = 1\)[/tex], [tex]\(y = 1\)[/tex]
- [-1, 1, 1]
3. When [tex]\(x_1 = 1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y = 1\)[/tex]
- [1, -1, 1]
4. When [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = -1\)[/tex], [tex]\(y = -1\)[/tex]
- [-1, -1, -1]
As a result, the data set with 2 binary features, [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex], and their corresponding class labels [tex]\(y\)[/tex] can be represented as follows:
[tex]\[ \begin{align*} \begin{bmatrix} 1 & 1 & 1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \\ -1 & -1 & -1 \\ \end{bmatrix} \][/tex]
This provides the detailed, step-by-step breakdown of how we arrive at the final data set:
[tex]\[ \begin{bmatrix} [1, 1, 1], \\ [-1, 1, 1], \\ [1, -1, 1], \\ [-1, -1, -1] \end{bmatrix} \][/tex]
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