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Why should I tied the inputs to ground? or the supply +? I just don't get it
Shouldn't I lift up all the input pins and inject a high or low to monitor the output of the chip?
An input left unconnected is likely to pick up ambient 60-cycle fields (or other magnetic noise). This is amplified internally, possibly causing spikes on the supply pins or misbehavior in other sections of the chip, etc.
Even in normal operation, digital chips of the earlier TTL family (as opposed to the low-power families) are known as needing precautions due to their switching higher currents.
To troubleshoot chips by themselves, it might pay to make your own testing rig. Arrange some dip switches around an IC socket, so you can install various IC's, then apply various input combinations, and observe the resulting outputs.
Be careful to drain off static charge at all times when working with IC's.