Was the colony of New Jersey a part York?

Yes

Let me explain:

The Duke of York received a grant for all the lands from the Delaware River to the Connecticut River. However, only a small portion of New Jersey was actually settled by New York colonists. In 1664, the English took control of New Jersey from the Dutch and divided the colony into two parts East and West Jersey. East Jersey was sold to a group of Quakers who established a religious colony. West Jersey was sold to a group of settlers who were interested in establishing a commercial colony. The two colonies were reunited in 1702 and became a royal colony.

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