1. **Implement XOR gate using NAND gates:** The XOR output is true when inputs differ. The XOR function can be implemented using NAND gates as: $$XOR(A,B) = NAND(NAND(A, NAND(A,B)), NAND(B, NAND(A,B)))$$ Step-by-step: 1. Compute $NAND(A,B)$. 2. Compute $NAND(A, NAND(A,B))$ and $NAND(B, NAND(A,B))$. 3. Compute $NAND$ of these two results to get XOR output. 2. **Design and implement a 1-bit comparator:** A 1-bit comparator outputs if two 1-bit inputs are equal. - Inputs: $A$, $B$ - Outputs: - $E$ = 1 if $A = B$ (Equality) - $G$ = 1 if $A > B$ - $L$ = 1 if $A < B$ Expressions: - $E = (A ext{ AND } B) ext{ OR } ( eg A ext{ AND } eg B)$ - $G = A ext{ AND } eg B$ - $L = eg A ext{ AND } B$ 3. **Cattle device logic system:** i. Truth table for three feeding stations $X$, $Y$, $Z$ where output $F=1$ when two or more stations are empty (assume empty= 1): |X|Y|Z|F| |-|-|-|-| |0|0|0|0| |0|0|1|0| |0|1|0|0| |0|1|1|1| |1|0|0|0| |1|0|1|1| |1|1|0|1| |1|1|1|1| ii. Unsimplified logic expression: $$F = XYZ + XY\overline{Z} + X\overline{Y}Z + \overline{X}YZ$$ iii. Using Karnaugh map simplification (for $X$, $Y$, $Z$), grouping yields: $$F = XY + YZ + XZ$$ iv. Logic diagram: Three AND gates with inputs $(X,Y)$, $(Y,Z)$, and $(X,Z)$ connected to an OR gate producing $F$. 4. **Circuit equivalence check:** Given: - Circuit 1 output $f = (a ext{ AND } b) ext{ OR } (b ext{ AND } c) ext{ OR } c$ - Circuit 2 output $g = (a ext{ AND } b) ext{ OR } (b ext{ AND } c)$ Check if $f = g$: - Notice $f = g + c$ Truth table for inputs $a,b,c$ and outputs $f,g$ shows $f eq g$ when $c=1$ and $a b=0$. Hence, the two circuits do **not** implement the same function.