Compatibility
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Compatibility between logic families

    Input type being driven
    3.3V CMOS 3.3V but 5V tolerant 5V HCT,  NMOS 5V LS  5V HC 5V 4000 CMOS
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3.3V CMOS YES YES YES YES Out of specification Out of specification
3.3V but 5V tolerant YES YES YES YES Open drain with resistor Open drain with resistor
5V HCT  Limiter needed YES YES YES YES YES
5V NMOS Limiter needed YES YES YES NEED PULL-UP RESISTOR NEED PULL-UP RESISTOR
5V LS Your mileage may vary YES YES YES NEED PULL-UP RESISTOR NEED PULL-UP RESISTOR
5V HC Limiter needed YES YES YES YES YES
5V 4000 CMOS Limiter needed YES YES Too weak  YES YES

Logic types

3.3V CMOS

Many microcontrollers are 3.3V CMOS devices, for example the LPC microcontrollers used in the MBED development tool.

HCMOS parts can be powered from 3.3V for compatibility with 3.3V microcontrollers.

3.3V but 5V tolerant

Some 3.3V microcontrollers feature pins that can accept 5V signals safely. There are limitations, typically the pin must be an input or an open-drain output requiring an external pull-up resistor.

5V HCT

A variant of HCMOS with lowered input threshold for compatibility with LS and NMOS

5V NMOS

NMOS devices such as early 1980s microprocessors and EPROMs

5V LS

The dominant logic family prior to HCMOS, common on "legacy" hardware. An enhanced version of TTL.

5V HC

HCMOS powered from 5V

5V 4000 CMOS

4000 series CMOS when powered from a 5V supply. This doesn't have as much output drive as HCMOS.

Conditions

Out of specification

This combination will typically work but does not meet published requirements for reliable operation.

Operating at low speed a 3.3V powered device will output almost exactly 3.3V and a 5V powered device will have an input threshold of 2.5V. At high speeds the device output may not reach 3.3V within the given transition time as the given timings are based on reaching 90% or 3.0V. The 5V device input may need to reach 70% or 3.5V to meet the worst case input threshold. 

In other words it will work on the test bench but do not do it in production.

Open drain with resistor

Some 3.3v devices can have outputs pulled up to 5V by an external resistor, allowing for direct connection to 5V circuits

Limiter needed

This combination will overdrive the input unless some means is found to restrict it. At low speeds a series resistor or voltage divider may suffice. At higher speeds a pass-transistor circuit could be used to limit the voltage applied to the 3.3V device

Your mileage may vary

In principle the LSTTL gate should not be able to drive the CMOS input above 3.3V but I do not believe this behavior is guaranteed. VOH (output high) has a typical value of 3.4V but doesn't have an upper limit. In practice the LSTTL output has to pass through two diode junctions setting a practical limit of 3.8V which ought to be low enough for most devices. Again YMMV.

NEED PULL-UP RESISTOR

The output is not guaranteed to reach the levels of the input being driven unless pulled up to 5V. 

Too Weak

The output of a 4000-series CMOS gate isn't strong enough to reliably drive a LSTTL input. Exceptions are the 4049 and 4050 buffers which have a stronger output than other devices.