explain whether the following given are closed under addition:
B={0, 1}
T={0, 4, 8, 12, 16, ...}
F={5, 6, 7, 8, 9, 10,...}
{x|x E W and x>100}
B clearly not, since 1+1=2, which is not in B
T clearly, since adding any two multiples of 4 yields another multiple of 4.
F yes, since adding any two integers yields another integer greater than 4.
No idea. What is W?
the "E" is one of the weird looking e's for math. The "W" is in italics.
If 4 plus 5 equals 9 is the set closed under whole numbers and addition?
B={0, 1}
Since 0 + 1 = 1, which is in the set B, B is closed under addition.
T={0, 4, 8, 12, 16, ...}
Let's check if this set is closed under addition. Adding any two numbers in this set will always give us a number that is a multiple of 4 (since all the numbers in the set have a common difference of 4). Therefore, T is closed under addition.
F={5, 6, 7, 8, 9, 10,...}
If we add any two numbers from this set, we might not always get a number in the set. For example, 5 + 9 = 14, which is not in the set F. So, F is not closed under addition.
{x|x E W and x>100}
This set consists of all natural numbers greater than 100. Adding any two numbers in this set will always give us a number greater than 100. Therefore, the set is closed under addition.
To determine whether a set is closed under addition, we need to verify if the sum of any two elements in the set produces another element that is also in the set.
1. Set B = {0, 1}:
- We can see that the sum of any two elements in B will always produce an element that is within B.
- For example, 0 + 0 = 0, 1 + 0 = 1, and 1 + 1 = 2 (not in B).
- Since the sum of any two elements in B does not always produce another element in B, set B is not closed under addition.
2. Set T = {0, 4, 8, 12, 16, ...}:
- To determine if set T is closed under addition, we need to verify the sum of any two elements.
- Let's choose two elements from T: 4 and 8.
- The sum of 4 and 8 is 12, which is in T.
- We can follow the same approach for any other two elements, and we will find that the sum always produces another element in T.
- Therefore, set T is closed under addition.
3. Set F = {5, 6, 7, 8, 9, 10, ...}:
- Again, we need to check if the sum of any two elements in F will result in another element within F.
- Let's take two elements from F: 6 and 7.
- The sum of 6 and 7 is 13, which is not in F.
- Therefore, set F is not closed under addition.
4. The set {x | x ∈ W and x > 100}:
- This set represents all the whole numbers greater than 100.
- To check if it is closed under addition, we need to verify if the sum of any two elements greater than 100 still gives a number greater than 100.
- Let's consider two elements from this set: 150 and 200.
- The sum of 150 and 200 is 350, which is still greater than 100.
- We can follow the same logic for any other two elements, and we will find that the sum always gives another number greater than 100.
- Therefore, the set {x | x ∈ W and x > 100} is closed under addition.
In summary:
- Set B = {0, 1} is not closed under addition.
- Set T = {0, 4, 8, 12, 16, ...} is closed under addition.
- Set F = {5, 6, 7, 8, 9, 10, ...} is not closed under addition.
- The set {x | x ∈ W and x > 100} is closed under addition.