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These three terms are interrelated, but nevertheless, there are some differences between them.
Partnership is an agreement between two parties working together to achieve a specific goal/outcome. At the same time, the goals of the partners are not necessarily the same.
Collaboration is one of the principles on which partnerships are built. You can also think of a collaboration between two parties as a partnership in which the parties have the same goals.
Relationship is the relationship of partners with each other, which does not necessarily imply partnership or joint work. One of the parties may simply provide services to the other party or carry out some assignments. In a global sense, relationships involve the desire to interact.
Membership is belonging to a community/organization. Most likely, membership can be seen as the result of a partnership.