Contents 1 Boolean Algebra F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 1 / 12 Boolean Algebra Section outline Boolean expression manipulation 1 Boolean Algebra Exclusive OR SOP from sets Series-parallel switching Boolean expressions circuits Functional completeness Shannon decomposition Distinct Boolean functions F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 2 / 12 1^2 A \ B (A \ B \ C) [ (A \ B \ C) abc + abc = ab 1^2^3^5 A (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) abc + abc + abc + abc = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1 A \ B \ C B 2 A \ B \ C 6 8 3 A \ B \ C 2 4 4 A \ B \ C 1 5 3 7 5 A \ B \ C A C 6 A \ B \ C 7 A \ B \ C 8 A \ B \ C F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 (A \ B \ C) [ (A \ B \ C) abc + abc = ab 1^2^3^5 A (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) abc + abc + abc + abc = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C B 2 A \ B \ C 6 8 3 A \ B \ C 2 4 4 A \ B \ C 1 5 3 7 5 A \ B \ C A C 6 A \ B \ C 7 A \ B \ C 8 A \ B \ C F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) [ (A \ B \ C) abc + abc + abc + abc = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C (A \ B \ C) [ B 2 A \ B \ C (A \ B \ C) 6 8 abc + abc = ab 3 A \ B \ C 2 4 ^ ^ ^ A 4 A \ B \ C 1 1 2 3 5 5 3 7 5 A \ B \ C A C 6 A \ B \ C 7 A \ B \ C 8 A \ B \ C F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 = ab + ab = a a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C (A \ B \ C) [ B 2 A \ B \ C (A \ B \ C) 6 8 abc + abc = ab 3 A \ B \ C 2 4 ^ ^ ^ A 4 A \ B \ C 1 1 2 3 5 5 3 7 (A \ B \ C) [ 5 A \ B \ C A C (A \ B \ C) [ 6 A \ B \ C (A \ B \ C) [ 7 A \ B \ C (A \ B \ C) 8 A \ B \ C abc + abc + abc + abc F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 a I have an item from A a I don’t have an item from A ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C (A \ B \ C) [ B 2 A \ B \ C (A \ B \ C) 6 8 abc + abc = ab 3 A \ B \ C 2 4 ^ ^ ^ A 4 A \ B \ C 1 1 2 3 5 5 3 7 (A \ B \ C) [ 5 A \ B \ C A C (A \ B \ C) [ 6 A \ B \ C (A \ B \ C) [ 7 A \ B \ C (A \ B \ C) 8 A \ B \ C abc + abc + abc + abc = ab + ab = a F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 ab + c I have an item from A but not from B or an item from C Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C (A \ B \ C) [ B 2 A \ B \ C (A \ B \ C) 6 8 abc + abc = ab 3 A \ B \ C 2 4 ^ ^ ^ A 4 A \ B \ C 1 1 2 3 5 5 3 7 (A \ B \ C) [ 5 A \ B \ C A C (A \ B \ C) [ 6 A \ B \ C (A \ B \ C) [ 7 A \ B \ C (A \ B \ C) 8 A \ B \ C abc + abc + abc + abc = ab + ab = a a I have an item from A F TECHN O OL E O T G U Y IT K a I don’t have an item from A T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 Boolean Algebra SOP from sets Sum of products from sets Selections Regions 1^2 A \ B 1 A \ B \ C (A \ B \ C) [ B 2 A \ B \ C (A \ B \ C) 6 8 abc + abc = ab 3 A \ B \ C 2 4 ^ ^ ^ A 4 A \ B \ C 1 1 2 3 5 5 3 7 (A \ B \ C) [ 5 A \ B \ C A C (A \ B \ C) [ 6 A \ B \ C (A \ B \ C) [ 7 A \ B \ C (A \ B \ C) 8 A \ B \ C abc + abc + abc + abc = ab + ab = a a I have an item from A F TECHN O OL E O T G U Y IT K a I don’t have an item from A T H S A N R I A N G A P I U D R N I 19 51 ab + c I have an item from A but not from B or an item from C yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 3 / 12 Such an expression is well formed or syntactically correct A fundamental product (FP) is a literal or a product of two or more literals arising from distinct variables A FP involving all the variables is a minterm – atoms in the BL A FP P1 is contained or included in P2 if P2 has all the literals of P1; then P2 ) P1 (P2 implies P1) A sum of products (SOP) expression is FP or a sum of two or more FPs P1;:::; Pn and 8i; j; Pi 6) Pj DeMorgan’s laws, distributivity, commutativity, idempotence, involution may be used to transform a Boolean expression to SOP A literal is a variable (a) or its complement (a) A Boolean expression is a string built from literals and the Boolean operators without violating their arity Grouping with parentheses is permitted Boolean Algebra Boolean expressions Boolean lattice (BL) for 2 variables 1 a + b a + b a + b a + b a b b ⊕ c a $ b b a ab ab ab ab 0 F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 4 / 12 Such an expression is well formed or syntactically correct A fundamental product (FP) is a literal or a product of two or more literals arising from distinct variables A FP involving all the variables is a minterm – atoms in the BL A FP P1 is contained or included in P2 if P2 has all the literals of P1; then P2 ) P1 (P2 implies P1) A sum of products (SOP) expression is FP or a sum of two or more FPs P1;:::; Pn and 8i; j; Pi 6) Pj DeMorgan’s laws, distributivity, commutativity, idempotence, involution may be used to transform a Boolean expression to SOP Boolean Algebra Boolean expressions Boolean lattice (BL) for 2 variables 1 a + b a + b a + b a + b a b b ⊕ c a $ b b a ab ab ab ab 0 A literal is a variable (a) or its complement (a) A Boolean expression is a string built from literals and the Boolean operators without violating their arity F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 Grouping with parentheses yog, km s kOflm^ is permitted Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 4 / 12 A fundamental product (FP) is a literal or a product of two or more literals arising from distinct variables A FP involving all the variables is a minterm – atoms in the BL A FP P1 is contained or included in P2 if P2 has all the literals of P1; then P2 ) P1 (P2 implies P1) A sum of products (SOP) expression is FP or a sum of two or more FPs P1;:::; Pn and 8i; j; Pi 6) Pj DeMorgan’s laws, distributivity, commutativity, idempotence, involution may be used to transform a Boolean expression to SOP Boolean Algebra Boolean expressions Boolean lattice (BL) for 2 variables 1 Such an expression is well formed or a + b syntactically correct a + b a + b a + b a b b ⊕ c a $ b b a ab ab ab ab 0 A literal is a variable (a) or its complement (a) A Boolean expression is a string built from literals and the Boolean operators without violating their arity F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 Grouping with parentheses yog, km s kOflm^ is permitted Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 4 / 12 A FP involving all the variables is a minterm – atoms in the BL A FP P1 is contained or included in P2 if P2 has all the literals of P1; then P2 ) P1 (P2 implies P1) A sum of products (SOP) expression is FP or a sum of two or more FPs P1;:::; Pn and 8i; j; Pi 6) Pj DeMorgan’s laws, distributivity, commutativity, idempotence, involution may be used to transform a Boolean expression to SOP Boolean Algebra Boolean expressions Boolean lattice (BL) for 2 variables 1 Such an expression is well formed or a + b syntactically correct a + b a + b a + b A fundamental product (FP) is a literal a b b ⊕ c or a product of two or more literals arising from distinct variables a $ b b a ab ab ab ab 0 A literal is a variable (a) or its complement (a) A Boolean expression is a string built from literals and the Boolean operators without violating their arity F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 Grouping with parentheses yog, km s kOflm^ is permitted Chittaranjan Mandal (IIT Kharagpur) SCLD January 28, 2020 4 / 12 A FP P1 is contained or included in P2 if P2 has all the literals of P1; then P2 ) P1 (P2 implies P1) A sum of products (SOP) expression is FP or a sum of two or more FPs P1;:::; Pn and 8i; j; Pi 6) Pj DeMorgan’s laws, distributivity, commutativity, idempotence, involution may be used to transform a Boolean expression to SOP Boolean Algebra Boolean expressions Boolean lattice (BL) for 2 variables 1 Such an expression is well formed or a + b syntactically correct a + b a + b a + b A fundamental