1.

To find the sets  X  and  Y  based on the given information, we need to interpret the provided

conditions step by step.  ### Given: 

1.  X ∪ Y = { 1 , 2 , 3 , 5 , 6 , 8 , 9 , 10 } 

2.  X ∩ Y = { 1 , 5 } 

3.  Y − X = { 2 , 6 , 9 , 10 }

### Step 1: Using  Y − X

From condition 3, we know that: Y − X = { 2 , 6 , 9 , 10 } 

This means that the elements  2 , 6 , 9 ,  and  10  are in  Y  but not in  X . Therefore, we can express  Y  as:

Y = ( Y − X ) ∪ ( X ∩ Y ) 

Since  X ∩ Y = { 1 , 5 } , we can write:

Y = { 2 , 6 , 9 , 10 } ∪ { 1 , 5 } = { 1 , 2 , 5 , 6 , 9 , 10 }

### Step 2: Using  X ∪ Y

Now, we know:

X ∪ Y = { 1 , 2 , 3 , 5 , 6 , 8 , 9 , 10 } 

And we have  Y = { 1 , 2 , 5 , 6 , 9 , 10 } . Now we can find  X :

X = ( X ∪ Y ) − Y 

This gives us:

X = { 1 , 2 , 3 , 5 , 6 , 8 , 9 , 10 } − { 1 , 2 , 5 , 6 , 9 , 10 } 

Calculating this, we get:

X = { 3 , 8 }

### Step 3: Verification  Now we need to verify if the derived sets satisfy all the conditions. 

- **Checking  X ∪ Y **:

X ∪ Y = { 3 , 8 } ∪ { 1 , 2 , 5 , 6 , 9 , 10 } = { 1 , 2 , 3 , 5 , 6 , 8 , 9 , 10 } (True)

- **Checking  X ∩ Y **:

X ∩ Y = { 3 , 8 } ∩ { 1 , 2 , 5 , 6 , 9 , 10 } = { 1 , 5 } (True)

- **Checking  Y − X **:

Y − X = { 1 , 2 , 5 , 6 , 9 , 10 } − { 3 , 8 } = { 2 , 6 , 9 , 10 } (True)

### Conclusion 

Thus, the sets  X  and  Y  are: 

X = { 3 , 8 } 

Y = { 1 , 2 , 5 , 6 , 9 , 10 }

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