Which ethernet pair is used for half-duplex signaling?
In full-duplex ethernet, 1,2 does TX; while 3,6 does RX.
In half-duplex, in theory, only one of them is needed - which one?
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Both are needed in half-duplex.
Duplex basically means: Two transmission channels, one for sending, one for receiving.
For Ethernet, full duplex means: TX and RX can happen at the same time.
For Ethernet, half duplex means: TX and RX do not happen at the same time, but still, being duplex, using separate channels.
This differs from the use of the word half duplex in other transmission schemes, like serial communications.
This has to do with the origins and definitions of Ethernet. Most of this goes back directly to what was possible 30 years ago, all 10base signals go back to 1981 at least. 100base was just an extension of that. Gigabit Ethernet changes this and does proper full duplex, sending and receiving on all lines simultaneously.
Now, speaking of oldstyle Ethernet, 10base2 etc: The protocols are hardware-independent. The same signal would be encoded on optical or electrical transmission channels. Back then, optical channels could not easily switch between sending and receiving. Also, early structured cabling Ethernet was connected on a hub (not switch), so CSMA-CD had to be implemented, meaning senders had to be able to listen for incoming transmissions (conflicts) during their own sending. And additionally, the early protocol stacks ran on CPUs so wimpy they could not calculate transmission and reception at the same time, giving you reason to drive half-duplex in an environment that was otherwise perfectly capable of full-duplex.