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 }