Come on, do not take a sledgehammer to crack a nut. There is a simple example:
Let the be such infinite set that , where is the i-th prime number and .
Then infinite family is an example satisfying the requirements. Because for every distinct and and there are infinitely many primes.
I read a problem like this somewhere.
since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.
I actually did a similar problem to this (as you described). The question was "Does there exists an uncountable family of subsets of a countable set such that the interesction of any two is finite?" The answer is yes. I gave two examples. I actually did it on the forum if you are interested. But! The OP said that the sets need to be subsets of and as your sets are not this doesn't work. I refuse to actually answer the question upon the grounds that the OP has NEVER shown any effort.