Siman 414

It has already been explained that regarding eruvei techumin—one may only establish an eruv with his knowledge, for perhaps he does not wish to establish an eruv at all, so that he does not lose the techum of his home, or he does not wish to establish an eruv in this direction but rather in another direction. However, the sages said: (page 82:) A person may establish an eruv on behalf of his minor son and daughter, even if they are not supported at his table, in order to train them in mitzvot, for one only establishes an eruv for the purpose of a mitzvah, and their knowledge is not required, as the obligation of chinuch rests upon the father. And even if they protested against him not to establish an eruv on their behalf, and even if they established an eruv in another direction—their knowledge is of no consequence, and we depend upon their father. And similarly, he may establish an eruv on behalf of his Canaanite manservant and maidservant.

But his adult son and daughter, even if they are supported at his table, and his Hebrew manservant and maidservant, and his wife—he may only establish an eruv on their behalf with their consent. Nevertheless, there is a distinction between them and others, for regarding others, it is impossible for him to establish an eruv for them until they explicitly agree. But here, even if they did not explicitly agree, but rather did not protest against him, and did not establish another eruv and remained silent—they fulfill their obligation with his eruv, for presumably they are pleased with what he did. However, if they protested against him or if they established an eruv for themselves—his eruv is not an eruv on their behalf.

A minor of six years—goes out with his mother's eruv, and all the more so one younger, and there is no need to set aside two meals for him. But after six—it is forbidden to take him outside the techum, for we may not actively cause them to perform a prohibition. However, a six-year-old who cannot be without his mother—she is compelled to take him with her. And if he is a precocious child and does not need his mother even being five years old—she should not take him with her. (And see Magen Avraham, subsection 7, and the main point is like the first resolution, see there)