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hw1_sol - ECE341 Homework No 1 Solution Problem No 1(a w y...

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ECE341 Homework No. 1 Solution Problem No. 1 (a) w y z y + z w.(y + z) wy wz wy + wz 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 The comparison of the 5 th and 8 th columns proves the equivalence of the two functions. (b) w y z y z w + yz w + y w + z (w + y).(w + z) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 The comparison of the 5 th and 8 th columns proves the equivalence of the two functions. Problem No. 2 (a) g v GGG GG = (g±̅ + g G±) GGGGGGGGGGGGG Basic XOR equation from XOR truth table = (g±̅) GGGGGG . (g De Morgan’s Law = (g G + ±). (g + ±̅) De Morgan’s Law = g G. g + g G. ±̅ + ±. g + ±. ±̅ = 0 + g G. ±̅ + g. ± + 0 = g. ± + g G. ±̅ (b) ² ³y z ´
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= (g. (G z ± ±±± ± )) ±± ±±±±± ± + (g̅ . (G z ±±± ±±± )) From basic XOR gate property = ²g. ³G z´µ + (x ±. (y. z + y ±. z ±)) Involution and property derived in part (a) = g. (G. ¶̅ + G ±.¶) + g̅. G. ¶ + g̅. G ±.¶̅
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