Definition of "equivalent"

For now to serve and to minister, servile and ministerial, are terms equivalent.

1692–1717, Robert South, Twelve Sermons Preached upon Several Occasions, 6th edition, volumes (please specify |volume=I to VI), London: […] J[ames] Bettenham, for Jonah Bowyer, […], published 1727

A doorknob of whatever roundish shape is effectively a continuum of levers, with the axis of the latching mechanism—known as the spindle—being the fulcrum about which the turning takes place. Applying a force tangential to the knob is essentially equivalent to applying one perpendicular to a radial line defining the lever.

2012 March, Henry Petroski, “Opening Doors”, in American Scientist‎, volume 100, number 2, pages 112–3

Finite sets A and B are equivalent sets only when n(A) = n(B) i.e., the number of elements in A and B are equal.

c. 2005, P N Gupta Kulbhushan, Comprehensive MCQ's in Mathematics, page 3

All enumerable sets are equivalent to each other, but not to any finite set.

1950, E. Kamke, Theory of Sets, page 16

Equivalent sets should, by rights, have the same "number" of elements. For this reason we sometimes say that equivalent sets have the same cardinality.

2000, N. L. Carothers, Real Analysis, page 18

The equivalence theorem: If both M is equivalent to a subset N1 of N and N is equivalent to a subset M1 of M, then the sets M and N are equivalent to each other.

2006, Joseph Breuer, Introduction to the Theory of Sets, page 41

He owned that, if the Test Act were repealed, the Protestants were entitled to an equivalent, and went so far as to suggest several equivalents.

1849, Thomas Babington Macaulay, chapter 7, in The History of England from the Accession of James II, volume I, London: Longman, Brown, Green, and Longmans

Our decision about energy will test the character of the American people and the ability of the President and the Congress to govern. This difficult effort will be the "moral equivalent of war" — except that we will be uniting our efforts to build and not destroy.

1977 April 18, Jimmy Carter, President's Address to the Nation on Proposed National Energy Policy