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I am looking for a device or circuit that outputs HIGH when a pushbutton has been pressed (if the button is pressed again it is supposed to still output HIGH), also, if possible, another pushbutton to make the output LOW.

Is this possible without an IC? How to do it?

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What you want is called a set/reset flipflop. Various flipflops have R and S inputs, but you can also make a primitive one out of two NAND gates. For example:

Follow the logic signals thru the circuit to see how it works and how it has memory.

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    \$\begingroup\$ And of course this can be implemented with discrete transistors or FETs or even even tubes if one is feeling particularly nostalgic. \$\endgroup\$ – Chris Stratton Dec 2 '12 at 5:26
  • \$\begingroup\$ I think I'll try this with discrete components. This or the thyristor variant. \$\endgroup\$ – Gunnish Dec 3 '12 at 12:40
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One device that may work (High only, no low) is a silicon-controlled rectifier or SCR. Once the SCR gate goes high with the pushbutton, the diode will conduct until a power cycle.

Is there any reason why you wouldn't use an RS flip-flop?

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Olin is right, you are probably looking for a S-R latch.

But if you want to do it without an IC, you'll need to search for circuits using its older name : bistable multivibrator.

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If you want achieve that without any semiconductor, you can use a relais and use a normally open contact to feed the coil as soon the relaiy switch on. You can put a normally closed contact in series to that circuit to allow the 'reset' function: ( image from this wikipedia article )

enter image description here

This is the oldest fashion you can create a st/reset flip flop :)

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This could be done with a thyristor like so:
thyristor

S1 could replaced with a push switch instead of a toggle switch so that the output is reset, when S2 is pressed.

SRC: http://www.mikroe.com/old/books/keu/06.htm

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