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4.1 Switching Algebra

Return. Next. 4.1 Switching Algebra. 1. Introduction. Logic circuits are classified into two types: Combinational: whose outputs depend only on its current inputs.

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4.1 Switching Algebra

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  1. Return Next 4.1Switching Algebra 1. Introduction • Logic circuits are classified into two types: • Combinational: whose outputs depend only on its current inputs. • Sequential: depend not only on the current inputs but also on the past sequence of inputs, possibly arbitrarily far back in time. • Analysis and Synthesis: • Analysis start with a logic diagram and proceed to a formal description of the function performed by that circuit, such as a truth table or a logic expression.

  2. A1 x=0 if x≠1 A1’ x=1 if x≠0 A2 if x=0, then x=1 A2’ if x=1, then x=0 A3 0·0=0 A3’ 1+1=1 A4 1·1=1 A4’ 0+0=0 A5 0·1=1·0=0 A5’ 1+0=0+1=1 Return Back Next 4.1Switching Algebra • Synthesis do the reverse, starting with a formal description and proceeding to a logic diagram. 2. Axioms

  3. Identities T1 x+0=x T1’ x·1=x T2 x+1=1 T2’ x·0=0 Null elements Idempotency T3 x+x=x T3’ x·x=x Involution T4 x =x T4’ Complements T5 x+x=1 T5’ x·x=0 Return Back Next 4.1Switching Algebra 3. Theorems with One Variable

  4. T6 x+y=y+x Commutativity T6’ x·y=y·x T7 (x+y)+z=x+(y+z) Associativity T7’ (x ·y) ·z=x·(y·z) T8 x·y+x·z=x·(y+z) Distributivity T8’ (x+y) ·(x+z)=x+y·z Return Back Next 4.1Switching Algebra 4. Theorems with multi-variable I

  5. x+x·y=x T9 Covering T9’ x·(x+y)=x T10 x·y+x·y=x Combining T10’ (x+y) ·(x+y)=x T11 x·y+x·z+y·z=x·y+x ·z Consensus T11’ (x+y) ·(x+z) ·(y+z)= (x+y) ·(x+z) Return Back Next 4.1Switching Algebra 4. Theorems with multi-variable II

  6. x+x+…+x=x T12 T12’ x·x ·…· x=x T13 x1 · x2 ·…· xn = x1+x2+…+xn T13’ x1+x2+…+xn = x1 · x2 ·…· xn T14 F(x1 , x2, … xn,+, ·)= F(x1 , x2, … xn, · ,+) T15 F(x1 , x2, … xn)=x1 ·F(1, x2, … xn)+ x1 ·F(0, x2, … xn) F(x1 , x2, … xn)=[x1+F(0, x2, … xn)]·[x1+F(1, x2, … xn)] T15’ Return Back Next 4.1Switching Algebra 4. Theorems with multi-variable III Generalized idempotency DeMorgan’s theorems Generalized DeMorgan’s theorems Shannon’s expansion theorems

  7. x x y Z=x·y y x x Z=x ·y y y Z=x+y • If F(w,x,y,z)=(w·x)+(x·y)+(w·(x+z )) Then according to T14 F(w,x,y,z)= Z=x+y (w+x) ·(x+y) ·(w+(x·z)) Return Back Next 4.1Switching Algebra • Examples • Equivalent circuits according to DeMorgan’s theorem T13

  8. x1 · x2 ·…· xn = x1+x2+…+xn x1+x2+…+xn = x1 · x2 ·…· xn Return Back Next 4.1Switching Algebra 5. Duality • Principle of Duality: Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and · and + are swapped throughout. • Examples (Covering) x+x·y=x x·(x+y)=x x·y+x·z+y·z=x·y+x ·z (Consensus) (x+y) ·(x+z) ·(y+z)= (x+y) ·(x+z) (DeMorgan’s theorems)

  9. Return Back Next 4.1Switching Algebra • Consider the following statement x+x·y=x (T9) x·x+y=x (According the principle of duality) x+y=x (According theorem T3’) How absurd it is! Where did we go wrong? The problem is in operator precedence. Actually x+x·y=x+(x·y) ∴ x·(x+y)=x Operator precedence: ( ), AND, OR

  10. Sum-of-products expression: is a logic sum of product terms. (e.g. z+w·x·y+x·y·z) • Product-of-sums expression : is a logic product of sum terms. [ e.g. z·(w+x+y)·(x+y+z) ] Return Back Next 4.1Switching Algebra 6. Standard Representations of Logic Functions • Truth table: The brute-force representation simply lists the output of the circuit for every possible input combination.

  11. Normal term: is a product or sum term in which no variable appears more than once. e.g. z, w·x·y, x·y·z, w+x+y, x+y+z Return Back Next 4.1Switching Algebra • Minterm: An n-variable minterm is a normal product term with n literals. There are 2n such product terms.. • Maxterm: An n-variable maxterm is a normal sum term with n literals. There are 2n such sum terms.

  12. Row x y z F minterm maxterm 0 1 2 3 4 5 6 7 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 x·y·z x·y·z x·y·z x·y·z x·y·z x·y·z x·y·z x·y·z x+y+z x+y+z x+y+z x+y+z x+y+z x+y+z x+y+z x+y+z Canonical Sum Canonical Product Return Back Next 4.1Switching Algebra Minterm or maxterm number

  13. Return Back Next 4.1Switching Algebra • Examples • Write the canonical sum and product for each of the following logic function: • Solution:

  14. Return Back 4.1Switching Algebra • We have now learned five possible representations for a combinational logic function: • A truth table • An algebraic sum of minterms, the canonical sum. • A minterm list using the ∑ notation. • An algebraic product of maxterms, the canonical product. • A maxterm list using the ∏ notation.

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