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I have made up the example below.

(1) A lot of workers at major grocery stores have been laid off recently. These stores encompass/include ABC Market and Save-On-Groceries.

Some of my non-native English speaking friends think both verbs "encompass and include" are correct. My other friends think only "encompass" is correct. Please help me. Thank you very much.

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Only 'include' is idiomatic. I think you can forget about 'encompass' in this context. It's more at home in literary or journalistic environments, and even there it's not an exact synonym for 'include'.

They propose six personality groups which encompass the majority of people.

CJ