May a child make a Berachah on a Pasul set of 4 species that their parents gave them to learn how to make the Berachah and do the Mitzvah?
The Shulchan Aruch (OC 657) rules that if a child knows how to wave the 4 species appropriately, his father must, because of the obligation of Chinuch -- education, arrange for him to have a set. The Biur Halacha (sv Katan) notes that this is true even if they are unable to recite the proper blessing. In sv Kedei, he says that "it is obvious that the set must be kosher".
The Shulchan Aruch (658:6) also rules (based on Sukkah 46b) that one should not give a child his set as a gift to use on the first day (when one must own the set he uses to fulfill the mitzva) before using it himself because a child cannot give away his own property (no Daat Makneh). As such it seems that even external invalidation are problematic for a child. However, Mishna Berura there (sk 28 and Shar HaTziyun sk 36) quotes a number of Rishonim who disagree and suggests that the Shulchan Aruch even agrees that external invalidations are acceptable for the obligation of Chinuch.
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