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10); that is, if s =⇒ t, if s is logically equivalent to s, and if t is logically equivalent to t, then s =⇒ t . It is quite clear that the new relation =⇒ on P is not only a preordering but in fact a partial ordering. (If [s] =⇒ [t] and [t] =⇒ [s], then s =⇒ t and t =⇒ s, so that s ⇐⇒ t. ) In a similar fashion the other logical connectives ∼ (“not”), ∧ (“and”), and ∨ (“or”) may be unambiguously extended to the set P of propositions. Then, under the partial ordering =⇒ , the proposition p ∧ q really is the infimum of the propositions p and q, and p ∨ q really is the supremum of p and q.

It is useful to know that the kernel of a linear transformation is always a vector subspace of its domain, that its image is a vector subspace of its codomain, and that a necessary and sufficient condition for a linear transformation to be injective is that its kernel contain only the zero vector. 9. Proposition. If T : V → W is a linear transformation between vector spaces, then ker T is a subspace of V and ran T is a subspace of W . 10. Proposition. A linear transformation T is injective if and only if ker T = {0}.

If S is a nonempty set, then P(S) (partially ordered by inclusion) is a distributive lattice. Of course, not every lattice is distributive. 15. Example. On the set S = {l, a, b, c, m} define a partial order ≤ so that l < a < m, l < b < m, l < c < m, and no two of the elements in the set {a, b, c} are comparable. Then (S, ≤) is a lattice but is not distributive. (Proof : a ∧ (b ∨ c) = a while (a ∧ b) ∨ (a ∧ c) = l. 6. 5. 1. Definition. A function f : L → M between lattices is a lattice homomorphism if f (x ∨ y) = f (x) ∨ f (y) and f (x ∧ y) = f (x) ∧ f (y) for all x, y ∈ L.

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### Companion to Real Analysis by John M. Erdman

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