Definition of "or"

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A quick definition clarification: Does the set $\{(x,y):x =0 \,\,\,\,\text{or} \,\,\,\,y=1 \}$ include the element $(0,1)$? (Sorry, English is not my first language, I get confused sometimes... Also sorry that this may not be a very good math question.) Thanks.

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Yes. In mathematical contexts, A or B means that A is true and B is false, or B is true and A is false, or that both are true.