second last diagram is the key

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+12 votes

- Express the function $f(x,y,z) = xy' + yz'$ with only one complement operation and one or more AND/OR operations. Draw the logic circuit implementing the expression obtained, using a single NOT gate and one or more AND/OR gates.
- Transform the following logic circuit (without expressing its switching function) into an equivalent logic circuit that employs only $6$ NAND gates each with $2$-inputs.

+18 votes

Best answer

$f(x,y,z)=xy'+yz' =xy'z'+xy'z+x'yz'+xyz'$

$f(x,y,z)=\sum_m(2,4,5,6)$

$K$-map

$y'z'$ | $y'z$ | $yz$ | $yz'$ | |
---|---|---|---|---|

$x'$ | $0$ | $0$ | $0$ | $1$ |

$x$ | $1$ | $1$ | $0$ | $1$ |

By pairing of $1's$, we get two pairs $(2,6),(4,5)$ resulting in same expression $F= xy'+yz'$

But by pairing of $0's$, we get two pairs $(0,1),(2,7)$, we get $F'= yz+x'y'$

Take complement, $F= \overline{(yz)}.(x+y)$

so we can implement the function with $1$ NOT , $1$ OR and $2$ AND gates.

For the second part , we need to implement given circuit using NANDs only.

so best way is to replace OR with Invert NAND, $A+B = \overline{(A'B')}$

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