Which concept is more primary in modern formal math: set, function or natural number?

My reasons to ask this question are the following:

1) formal definition of a function is "a subset of cartesian product of two sets" so it is defined with the set and natrual number as basic concepts.

2) in axiomatic set theory we need to be able to use logical stetements, so we need to be able to say that the current statement is true or false. As i understand we need a set $\{ 0, 1\}$ or $\{ T, F \}$ or some other set (SET!) consisting of two (NUMBER!) different elements, one of which is considered to represent truth value, the other - false value.

3) some formal definitions of natural numbers which i met were made in set theory (so we need set as a basic concept) and when we want to speak about algebraic operations we need a function concept.

I've just began to learn foundations, i.e. logic, set theory and so on. So i apologize for the question which many of you may consider to be stupid. And i'll be very grateful for any answers and ideas. Thanks in advance!