If `A={x:x^(2)-2x+2gt0}andB={x:x^(2)-4x+3le0}`
A - B equals

Options

A. `(-oo,oo)`
B. (1,3)
C. `(3,oo)`
D. `(-oo,1)uu(3,oo)`

Explanation

To solve the problem, we need to find the sets A and B based on the given inequalities and then compute the difference A - B.

### Step 1: Determine Set A

We start with the inequality for set A:

\[ A = \{ x : x^2 - 2x + 2 > 0 \} \]

To analyze this inequality, we can rewrite it:

\[ x^2 - 2x + 2 = (x - 1)^2 + 1 > 0 \]

The expression \((x - 1)^2\) is always non-negative (greater than or equal to 0), and since we add 1 to it, the entire expression is always greater than 0 for all real numbers \(x\).

Thus, we conclude:

\[ A = (-\infty, \infty) \]

### Step 2: Determine Set B

Next, we consider the inequality for set B:

\[ B = \{ x : x^2 - 4x + 3 \leq 0 \} \]

We can factor this quadratic expression:

\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]

Now we need to find the values of \(x\) for which this product is less than or equal to 0. The roots of the equation are \(x = 1\) and \(x = 3\).

To determine the intervals where the product is non-positive, we can test the intervals defined by the roots:

1. For \(x < 1\): Choose \(x = 0\) → \((0 - 1)(0 - 3) = 3 > 0\)
2. For \(1 \leq x \leq 3\): Choose \(x = 2\) → \((2 - 1)(2 - 3) = -1 \leq 0\)
3. For \(x > 3\): Choose \(x = 4\) → \((4 - 1)(4 - 3) = 3 > 0\)

Thus, the solution to the inequality is:

\[ B = [1, 3] \]

### Step 3: Calculate A - B

Now we need to find the difference \(A - B\):

\[ A - B = (-\infty, \infty) - [1, 3] \]

When we subtract the interval \([1, 3]\) from \((- \infty, \infty)\), we are removing the values from 1 to 3. This results in two separate intervals:

1. From \(-\infty\) to \(1\) (not including \(1\))
2. From \(3\) to \(\infty\) (not including \(3\))

Thus, we can express the result as:

\[ A - B = (-\infty, 1) \cup (3, \infty) \]

### Final Answer

The final answer is:

\[ A - B = (-\infty, 1) \cup (3, \infty) \]