The complete 5.4 kton MINOS far detector has been taking data since the beginning of August 2003 at a depth of 2070 meters water-equivalent in the Soudan mine, Minnesota. This paper presents the first MINOS observations of muon neutrino and muon anti-neutrino charged-current atmospheric neutrino interactions based on an exposure of 418 days. The ratio of upward to downward-going events in the data is compared to the Monte Carlo expectation in the absence of neutrino oscillations giving: R_data(up/down)/R_MC(up/down) = 0.62^{+0.19}_{-0.14} (stat.) +- 0.02 (sys.). An extended maximum likelihood analysis of the observed L/E distributions excludes the null hypothesis of no neutrino oscillations at the 98 % confidence level. Using the curvature of the observed muons in the 1.3 T MINOS magnetic field muon neutrino and muon anti-neutrino interactions are separated. The ratio of muon neutrino to muon anti-neutrino events in the data is compared to the Monte Carlo expectation assuming neutrinos and anti-neutrinos oscillate in same manner giving: R_data(numubar/numu)/R_MC(numubar/numu) = 0.96^{+0.38}_{-0.27} (stat.) +- 0.15 (sys.), where the errors are the statistical and systematic uncertainties. Although the statistics are limited, this is the first direct observation of atmospheric neutrino interactions separately for muon neutrinos and muon anti-neutrinos. ; Comment: 17 pages, 16 figures, submitted to Phys. Rev. D