Prove the following rules using the proof by perfect induction.1. XY�

1.	Prove the following rules using the proof by perfect induction.1. XY’ + XY = X2. X+Y=Y+X
Answer» 1. XY’ + XY = X By proof of perfect induction X = 0, Y =1 XY’ + XY = X LHS = 0.0 + 0.1 = 0 + 0 = 0 By proof of perfect induction X = 1, Y = 0 LHS = 1.1 + 1.0 = 1 + 0 = 1. 2. X + Y = Y + X By proof of perfect induction X = 0, Y = 1 LHS = 0 + 1 = 1 RHS = 1 + 0=1 LHS = RHS By proof of perfect induction X = 1, Y = 0 LHS = 1 + 0 = 1 RHS = 0 + 1 = 1 LHS = RHS

Prove the following rules using the proof by perfect induction.1. XY’ + XY = X2. X+Y=Y+X

Answer»

1. XY’ + XY = X
By proof of perfect induction X = 0, Y =1
XY’ + XY = X
LHS = 0.0 + 0.1 = 0 + 0 = 0
By proof of perfect induction X = 1, Y = 0
LHS = 1.1 + 1.0 = 1 + 0 = 1.

2. X + Y = Y + X
By proof of perfect induction X = 0, Y = 1
LHS = 0 + 1 = 1
RHS = 1 + 0=1
LHS = RHS
By proof of perfect induction X = 1, Y = 0
LHS = 1 + 0 = 1
RHS = 0 + 1 = 1
LHS = RHS

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Prove the following rules using the proof by perfect induction.1. XY’ + XY = X2. X+Y=Y+X

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