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Einzig vs. einsam

While the German words einzig and einsam both stem from the number eins (one), they have completely different meanings. Einzig relates to quantity and uniqueness, whereas einsam describes an emotional state of loneliness.

Einzig

A2
The word einzig means only, sole, or unique. It is a neutral term used to describe something that is the single one in existence or in a specific context. It refers to a factual quantity or a state of being one-of-a-kind, carrying no emotional weight.
Das ist meine einzige Chance.
(This is my only chance.)
Er war der einzige Gast auf der Party.
(He was the only guest at the party.)
Sein einziges Ziel ist es, die Prüfung zu bestehen.
(His sole objective is to pass the exam.)
Diese Methode ist in ihrer Art einzig.
(This method is unique in its kind.)
Das ist einzig und allein deine Verantwortung.
(That is solely your responsibility.)

Einsam

A2
The word einsam means lonely or lonesome. It describes the negative feeling of being socially or emotionally isolated, regardless of whether one is physically alone or not. It is purely about an emotional state.
Nach dem Umzug in die neue Stadt fühlte er sich sehr einsam.
(After moving to the new city, he felt very lonely.)
Das alte Haus steht einsam auf dem Hügel.
(The old house stands lonesomely on the hill.)
Man kann sich auch in einer großen Menschenmenge einsam fühlen.
(You can also feel lonely in a large crowd.)
Es war ein langer und einsamer Winter für sie.
(It was a long and lonesome winter for her.)
Ein Spaziergang an einem einsamen Strand kann sehr friedlich sein.
(A walk on a deserted beach can be very peaceful.)

Summary

In essence, einzig is a word of fact, meaning only or unique. You can be the einzige (only) person in a room. Einsam, on the other hand, is a word of feeling, meaning lonely. You can be in a room full of people and still feel einsam (lonely). The first describes a number, the second describes an emotion.