Are the set of all square numbers under 50 and the set of all triangular numbers that are less than 50 disjoint?

No, the two sets are not disjoint.

Step-by-step explanation:

Set of triangular numbers to 50: 1, 3, 6, 10, 15, 21, 28, 36, 45

Set of square numbers to 50: 1, 4, 9, 16, 25, 36, 49

They have 1 and 36 in common, so they are not disjoint, or in other words, they have some of the same numbers.