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I'm doing a full-adder using logic gates and I want to express the carry bit with a gate.

So I found the minterms of the carry bit

x'yz + xy'z + xyz' + xyz

Is there a 3-input gate that does that expression just like the 3 way XOR?

for example:

x XOR y XOR z = x'y'z + x'yz' + xy'z' + xyz

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  • \$\begingroup\$ Its not clear what you mean by this question. For anything other than some simple 2-input gates, there are many possibilities to generate compound gates. In ASIC libraries, these are common. In FPGA, the implementation doesn't target gates, it targets lookup tables. \$\endgroup\$ May 19, 2016 at 11:28

3 Answers 3

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You can use a K-map to reduce the number of logic gates needed. It helps to draw out the truth table for the MSB (carry - out) to fill in the K-Map.

The beauty of the K-map is that circling neighboring squares allows you to simplify the Boolean expression. You see there are 3 pairs circled. Then how many gates would you need?

Carry Out K-Map

Another hint to remember is that you can turn sum of products logic to NAND-NAND logic further reducing the types of gates needed.

I have never seen a 3 input Majority Gate. I am answering the question in terms of gates I know exist such as AND, OR, NOT and NAND gates.

I am wondering if you are using 7400 TTL gates. If that is the case, I don't think a 3 Input Majority gate exists.

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  • \$\begingroup\$ That's very nice. What's your answer to the question? \$\endgroup\$
    – pipe
    May 19, 2016 at 19:47
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Yes, it's called a 3-input "majority" gate, or MAJ3 in some libraries. You won't find one as a discrete device, but many FPGA libraries include such a function.

You do realize that the fourth term in your Boolean expression is redundant, right?

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Note that...

x'yz + xy'z + xyz' + xyz

reduces to a simpler form

xy + xz + yz

As can be seen from the truth table.

XYZ
000 | 0
001 | 0
010 | 0
011 | 1
100 | 0
101 | 1
110 | 1
111 | 1

This is a majority gate.

If you are buying chips to make this then you may use a three input and gate and three or gates.

Dual 3 input AND gate SN74HC11N

DATASHEET: http://www.ti.com/lit/ds/symlink/sn74hc11.pdf (Datasheet)

http://www.digikey.com/product-detail/en/texas-instruments/SN74HC11N/296-8217-5-ND/376871 (Buy it on Digikey for $0.50)

Quad 2-input OR gate SN74HC32N

http://www.ti.com/lit/ds/symlink/sn74hc32.pdf (Datasheet)

http://www.digikey.com/product-detail/en/texas-instruments/SN74HC32N/296-1589-5-ND/277235 (Buy it on Digikey for $0.46)

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