A set of n elements has 2ⁿ subsets. One of these subsets is the set itself, so this set would have 2ⁿ - 1 proper subsets.
The given set has 4 elements, so it has 2⁴ = 16 subsets and 2⁴ - 1 = 15 proper subsets.
Just to illustrate the claim, we have
• subsets of size 0 (1 of these):
{ } (empty set)
• subsets of size 1 (4 of these):
{1/6}, {1/7}, {1/8}, {1/9}
• subsets of size 2 (6 of these):
{1/6, 1/7}, {1/6, 1/8}, {1/6, 1/9}, {1/7, 1/8}, {1/7, 1/9}, {1/8, 1/9}
• subsets of size 3 (4 of these):
{1/6, 1/7, 1/8}, {1/6, 1/7, 1/9}, {1/6, 1/8, 1/9}, {1/7, 1/8, 1/9}
• subsets of size 4 (1 of these):
{1/6, 1/7, 1/8, 1/9}
and the last one is the only subset that is not a proper subset, because that subset = the set.
