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My favorite short story in this collection is Black Destroyer, written by A. E. Van Gogt. It is the story of the last survivor of a dying race, whose only chance for survival comes in the form of an Earth ship. Other good stories include the humorous Trouble With Water by H. L. Gold and I, Robot by Eando Binder (no relation to Asimov's own book by the same name).
This is the first book in a series of classic science fiction collections by Isaac Asimov, and is well worth your interest.
Complete listing of Short Stories in this collection:
I, Robot -- Eando Binder
The Strange Flight of Richard Clayton -- Robtert Bloch
Trouble with Water -- H.L. Gold
Cloak of Aesir -- Don A. Stuart
The Day is Done -- Lester Del Rey
The Ultimate Catalyst -- John Taine
The Gnarly Man -- John Taine
Black Destroyer -- A.E. Van Gogt
Greater Than Gods -- C.L. Moore
Trends -- Isaac Asimov
The Blue Giraffe -- L. Sprague De Camp
The Misguided Halo -- Henry Kuttner
Heavy Planet -- Milton A. Rothman
Life-Line -- Robert A. Heinlein (his first)
Ether Breather -- Theodore Sturgeon
Pilgrimage -- Nelson Bond
Rust -- Joseph E. Kelleam
The Four-Sided Triangle -- William F. Temple
Star Bright -- Jack Williamson
Misfit -- Robert A. Heinlein (his second)
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FAR CENTAURUS A. E. Van Vogt
DEADLINE Cleve Cartmill
VEIL OF ASTELLAR Leigh Brackett
SANITY Fritz Leiber
INVARIANT John R. Pierce
CITY Clifford D. Simak
ARENA Fredric Brown
HUDDLING PLACE Clifford D. Simak
KINDNESS Lester Del Rey
DESERTION Clifford D. Simak
WHEN THE BOUGH BREAKS Kuttner & Moore
KILLDOZER! Theodore Sturgeon
NO WOMAN BORN C. L. Moore
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