Let ℱ be the set of all functions from {1, … , 𝑛} to {0,1}. Define the binary…

Concept

A relation ≼ on a set S is a partial order when it is reflexive, antisymmetric, and transitive; the pair (S, ≼) is then a poset. A poset (S, ≼) is a lattice when every pair a, b ∈ S has both a meet a∧b (greatest lower bound) and a join a∨b (least upper bound) that lie in S. An equivalence relation additionally requires symmetry — a genuine order relation with more than one comparable pair can never also be symmetric, since symmetry together with antisymmetry would force every comparable pair to be equal.

Application to this relation

Each function f ∈ ℱ can be read as a bit-string of length n (its values at 1,…,n), and f ≼ g compares these bit-strings coordinate by coordinate — this is exactly the product (coordinate-wise) order on {0,1}ⁿ.

1. Reflexive — f(x) ≤ f(x) holds at every x, so f ≼ f for every f.

2. Antisymmetric — if f ≼ g and g ≼ f then f(x) ≤ g(x) and g(x) ≤ f(x) at every x, forcing f(x) = g(x) everywhere, so f = g.

3. Transitive — if f ≼ g and g ≼ h then f(x) ≤ g(x) ≤ h(x) at every x, so f ≼ h.

4. Symmetric? — Take f with f(1)=0 and g with g(1)=1, agreeing elsewhere: f ≼ g holds, but g ≼ f fails at x=1. So ≼ is NOT symmetric, and therefore NOT an equivalence relation either.

5. Lattice? — Define (f∧g)(x) = min(f(x), g(x)) and (f∨g)(x) = max(f(x), g(x)); both are functions in ℱ and are respectively the greatest lower bound and least upper bound of f and g under ≼. Every pair has both, so (ℱ, ≼) is a lattice.

Cross-check with n = 2

Represent the four functions as bit-strings 00, 01, 10, 11 (values at x=1,2). Then 00 ≼ 01 ≼ 11 and 00 ≼ 10 ≼ 11, while 01 and 10 are incomparable (01 ⋠ 10 and 10 ⋠ 01) — exactly the diamond-shaped Hasse diagram of the Boolean lattice B₂. Their meet is 00 (coordinate-wise min) and their join is 11 (coordinate-wise max), both present in the set, confirming the lattice property independently of the general argument above; the incomparability of 01 and 10 again confirms ≼ is not symmetric (it is not even a total order).

Conclusion

(ℱ, ≼) is a partial order and (ℱ, ≼) is a lattice; the relation is not symmetric, so it is not an equivalence relation.