Let A be a subset of T={1,2,…,100} subject to the condition that if 2 distinct elements a and b are in A, then a+b is not in A. What is the maximum value of |A|, which is the number of elements in A?
To find the maximum value of |A|, we need to find a subset A of T={1,2,...,100} that satisfies the given condition and has the largest number of elements.
Since A cannot contain any two distinct elements a and b, where a+b is also in A, we can start by considering the smallest elements of T.
Let's start with the number 1. We can include it in A, as there is no other number we need to exclude.
Next, we consider the number 2. Since 2+1=3, and 3 is not in A, we can include 2 in A.
Continuing this pattern, we can include the numbers 4, 8, 16, 32, and 64, as none of them can be expressed as the sum of two distinct elements already in A.
However, when we consider the number 3, we realize that 3+2=5, and 5 is not in A. Therefore, we need to exclude 3 from A.
By following this approach, we can include the numbers 1, 2, 4, 8, 16, 32, and 64 in A, giving us a total of 7 elements. None of these elements can be paired to form a sum that is already in A.
So, the maximum value of |A| is 7.