In ordinary set theory, an element is a member of the union of two sets if it belongs to either of them.  In other words, the only elements that are not in the union are those that aren't in either set.  Therefore, if something has a positive degree of membership in either of two fuzzy sets, it should have a positive degree of membership in their union.  A similar chain of reasoning shows us that the intersection of two fuzzy sets should consist of the elements that have a positive degree of membership in both sets.

That's all well and good, but there are other properties that we'd like to have as well.  The major relationships between the set operations in the case of ordinary sets are De Morgan's laws, which state that (A∩B)c=Ac∪Bc(A∩B)c=Ac∪Bc and (A∪B)c=Ac∩Bc(A∪B)c=Ac∩Bc.  If we take the degree of membership of an element in AcAc to be one minus its degree of membership AA, then we can get De Morgan's laws to hold by defining the degree of membership in the union of two fuzzy sets to be the maximum of the degree of membership in either set, and the degree of membership in the intersection to be the minimum of the original degrees of membership.  I understand that there are other options, but this seems to be pretty standard.

The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form

i:[0,1]×[0,1] → [0,1].For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].

Axioms for fuzzy intersection[edit]

Axiom i1. Boundary conditioni(a, 1) = aAxiom i2. Monotonicityb ≤ d implies i(a, b) ≤ i(a, d)Axiom i3. Commutativityi(a, b) = i(b, a)Axiom i4. Associativityi(a, i(b, d)) = i(i(a, b), d)Axiom i5. Continuityi is a continuous functionAxiom i6. Subidempotencyi(a, a) ≤ aAxiom i7. Strict monotonicityi (a1, b1) ≤ i (a2, b2) if a1 ≤ a2 and b1 ≤ b2

Axioms i1 up to i4 define a t-norm (aka fuzzy intersection). The standard t-norm min is the only idempotent t-norm (i. e. i (a1, a1) = a for all a ∈ [0,1])