7 proves this theorem by the method of perfect

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Unformatted text preview: imilarly, Figure 6.6 proves Theorem 3(b) by the method of perfect induction. X y x-y x+x■y 0 0 1 1 0 l 0 I 0 0 0 1 0 0 1 1 Figure 6.5. Truth table for proving Theorem 3(a) by the method of perfect-induction. y X 0 0 1 1 x+y x ■ (x + y) 0 l 0 l 0 1 1 1 0 0 1 1 Figure 6.6. Truth table for proving Theorem 3(b) by the method of perfect induction. Theorem 4 (Involution Law) x=x Proof Figure 6.7 proves this theorem by the method of perfect induction. X X X 0 1 1 0 i Note that Theorem 4 has no dual since it deals with the NOT operator which is unary operator. Theorem 5 (a) x-(x +y) = x-y (b) x + x-y = x + y Proof of (a) Figure 6.8 proves this theorem by the method of perfect induction. X y X x+y x-(x + y) x-y 0 0 1 1 0 l 0 1 1 1 0 0 1 1 0 1 0 0 0 1 _j 0 0 0 1 Figure 6.8. Truth table for proving Theorem 5(a) by the method of perfect induction. Proof of (b) Figure 6.9 proves this theorem by the method of perfect induction. X y X X- y x+ \ • y x+y 0 0 1 1 0 l 0 l 1 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 Figure 6.9. Truth table for proving Theorem 5(b) by the method of perfect induction. Theorem 6 (De Morgan's Law) (a) x+y = x • y (b) x • y = x + y Proof of (a) ' Figure 6.10 proves this theorem by the method of perfect induction. X y x+y X X x+y y y 0 0 1 1 0 l 0 l 0 1 1 1 1 0 0 0 1 1 0 0 l 0 l 0 1 0 0 0 Figure 6.10. Truth table for proving Theorem 6(a) by the method of perfect induction. Proof of (b) Figure 6.11 proves this theorem by the method of perfect induction. X y x • y- 0 0 1 1 0 0 0 0 1 I 0 I x-y '-y 1 1 1 0 X' y x+y 1 1 0 0 l 0 l 0 1 1 1 0 Figure 6.11. Truth table for proving Theorem 6(b) by the method of perfect induction. Theorems 6(a) and 6(b) are important and very useful. They are known as De Morgan's laws. They can be extended to n variables as given below: X1+X2+X3+... + Xn = X1-X2-X3-...-X11 x,.x2-x3.... xn=x1+x2 The basic Boolean identities are summarized in Figure 6.12. It is suggested that the readers should become well conversant with the identities given in this table in order to use the algebr...
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