 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. 4.1 Switching Algebra ReturnNext 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.

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

4.1 Switching Algebra NextBackReturn 3. Theorems with One Variable x · x=0T5’x+x=1T5 T4’x =xT4 x · x=xT3’x+x=xT3 x · 0=0T2’x+1=1T2 x · 1=xT1’x+0=xT1 Identities Null elements Idempotency Complements Involution

4.1 Switching Algebra NextBackReturn 4. Theorems with multi-variable I x · y+x · z=x · (y+z)T8 (x+y)+z=x+(y+z)T7 x+y=y+xT6 (x+y) · (x+z)=x+y · zT8’ (x · y) · z=x · (y · z)T7’ x · y=y · xT6’ Commutativity Associativity Distributivity

4.1 Switching Algebra NextBackReturn (x+y) · (x+z) · (y+z)= (x+y) · (x+z) (x+y) · (x+y)=xT10’ x · y+x · y=xT10 x · (x+y)=xT9’ x+x · y=x T9 Combining Covering T11’ T11 Consensus x · y+x · z+y · z=x · y+x · z 4. Theorems with multi-variable II

4.1 Switching Algebra NextBackReturn 4. Theorems with multi-variable III F(x 1, x 2, … x n )=x 1 · F(1, x 2, … x n )+ x 1 · F(0, x 2, … x n ) F(x 1, x 2, … x n,+, · )= F(x 1, x 2, … x n, ·,+) x 1 +x 2 + … +x n = x 1 · x 2 · … · x n T13’ x 1 · x 2 · … · x n = x 1 +x 2 + … +x n T13 x · x ·… · x=xT12’ x+x+ … +x=x T12 T15 T14 T15’ F(x 1, x 2, … x n )=[x 1 + F(0, x 2, … x n )] ·[x 1 + F(1, x 2, … x n )] Generalized idempotency DeMorgan ’ s theorems Generalized DeMorgan ’ s theorems Shannon ’ s expansion theorems

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

4.1 Switching Algebra NextBackReturn 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 x+x · y=x x · (x+y)=x (Covering) x · y+x · z+y · z=x · y+x · z (x+y) · (x+z) · (y+z)= (x+y) · (x+z) (Consensus) x 1 · x 2 · … · x n = x 1 +x 2 + … +x n x 1 +x 2 + … +x n = x 1 · x 2 · … · x n (DeMorgan ’ s theorems)

4.1 Switching Algebra NextBackReturn 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

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

4.1 Switching Algebra NextBackReturn  Minterm: An n -variable minterm is a normal product term with n literals. There are 2 n such product terms..  Maxterm: An n -variable maxterm is a normal sum term with n literals. There are 2 n such sum terms. 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

4.1 Switching Algebra NextBackReturn x y z x+y+z x·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·zx·y·z maxtermmintermFRow Minterm or maxterm number Canonical SumCanonical Product

4.1 Switching Algebra NextBackReturn Examples  Write the canonical sum and product for each of the following logic function:  Solution:

A truth table 4.1 Switching Algebra We have now learned five possible representations for a combinational logic function: BackReturn 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.