Why is a women allowed to make a berachoh on mitzvot they are not obligated to do (like lulav and etrog for example) if in the berachah we recite "vetzivanu"- and has commanded us if we were not commanded? It seems as if we are lying if we were not commanded?

This is a good and ancient question The Rambam (Tzitzit 3, 9) writes that women should not say the Brachah for Mitzvot they are not oblige to do as they can’t say “Vetzivanu” – and he had commanded us; Rabenu Tam holds that they can say the Brachah as for the word Vetzivanu – that refers to the men and the Jewish people as a whole. The Shulchan Aruch (OC 589, 6) holds like the Rambam, which is the standard Sephardic custom; and the Rema (OC 17, 2) like Rabenu Tam, which is the standard Ashkenazi custom.