We known that Z={...,−3,−2,−1,0,1,2,3,...}\Z = \{ ... , - 3,-2,-1,0,1,2,3, ... \}Z={...,−3,−2,−1,0,1,2,3,...} .
Hence, 3Z=3×{...,−3,−2,−1,0,1,2,3,...}={...,−9,−6,−3,0,3,6,9,...}3\Z = 3 \times\{ ... , - 3,-2,-1,0,1,2,3, ... \} = \{ ... , - 9,-6,-3,0,3,6,9, ... \}3Z=3×{...,−3,−2,−1,0,1,2,3,...}={...,−9,−6,−3,0,3,6,9,...} .
By completeness property of real numbers, for every k∈Rk \in \Rk∈R , ∃ n∈Z:n≥k\exist \ n \in \Z : n \geq k∃ n∈Z:n≥k . Hence, Z\ZZ is an infinite set.
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