Technically speaking, all three notes of a triad need to be present for you to have a triad. However, certain combinations of notes can imply the sound of a chord. So for example, if the notes C and E are played together, they may imply a C major chord. This will ultimately depend upon context.

Also, when you play one note, you are actually hearing that one note (or fundamental note) and you are also hearing a combination of different overtones. The different overtones that are present are what characterize the unique sound of each instrument. It’s the presence or absence of certain overtones from the harmonic overtone series that allow us to hear a difference between the violin and the guitar even when they play the exact same note.

The reason some teachers will show beginners simple two note voicings for chords is to simplify things for the beginner.

Single note lines may imply a chord when they outline one in the melody. Otherwise one note by itself is generally not enough to imply a definite or even implied harmony. This is because there are many different chords that a single note can be part of. The note C can be the root of the C major chord (C-E-G); the b3 of an A minor chord (A-C-E); or the fifth of an F major chord (F-A-C).

I hope this clarifies things. Let me know.