CH3 - BOOLEAN ALGEBRA Continued Hasan Zidan Department of...

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BOOLEAN ALGEBRA ontinued Continued Hasan Zidan epartment of Electrical Engineering Department of Electrical Engineering AUST 1
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Exclusive-OR function One gate type is the XOR Gate (Exclusive OR gate). OR A B F = A B A B Y 0 0 0 XOR 0 1 1 1 0 1 1 0 XOR gate is common in logic circuits that do binary addition/subtraction. 1 1 0 y Note that: = A F A B = A B + AB (A ) (AB 2 = + B) (AB )
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Equivalence Function quivalence is the complement of OR Equivalence is the complement of XOR . A F = (A B) ’= A B B Y B A B Y 0 0 1 0 1 0 A 1 0 0 1 1 1 -- alternate symbol B B Note: In order to simplify expressions containing AND and OR as ell as XOR and t is usually desirable to replacing XOR and well as XOR and , it is usually desirable to replacing XOR and in terms of AND and OR. 3
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CONSENSUS THEOREM Consensus theorem states: XY + X’Z + YZ = XY + X’Z The YZ term is called the consensus term and is redundant. The consensus term is formed from a PAIR OF TERMS in which a variable ( X ) and its complement ( X’ ) are present ; the consensus term is formed by p ;y multiplying the two terms and leaving out the selected variable and its complement. The consensus of XY , X’Z is YZ . 4
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Prove the CONSENSUS THEOREM Consensus Theorem Proof: XY + X’Z + YZ = XY + X’Z + (X + X’) YZ = XY + X’Z + XYZ + X’YZ = (XY + XYZ) + (X’Z + X’YZ) = XY (1 + Z) + X’Z (1 + Y) Y + X’Z = XY + X Z You could also use a truth table to prove this.
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CH3 - BOOLEAN ALGEBRA Continued Hasan Zidan Department of...

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