Example Simplify Answer 1. (x + y) (x + z) (y + z) (x + y).(x + z) 2. xy + xz + yzw Xy + yz zw + xz 3. A + B +CD (A + B) . (C + D) 4. ABC + ABC + ABC Connot be simplicied further 5. xy + xy + xy+xyz + xyz X + y in example 4 above, all the three variable are present in each of four Products (calling AND operation Product). Such expression in which all the variables (either complemented or otherwise) are connected with AND operation in each term and all terms are connected with OR operator, are said to be expression in sop sum of Product term. Each term of SOP form is called minterm. The dual of SOP is and Dual of minterm is a maxterm. POS (Product of sums) Example: (x,y,z)= xy is not in SOP form because variable z is not present in the product. To convert it is SOP form multiply the term with (z+z)=1 F(x,y,z)=xyz+xy,z Now both terms are minterm and f is in SOP form. IMPLEMENTATIONS OF BOOLEAN ALGERA IN COMPUTER Logic circuits Logic Circuits are used to implement Boolean Algebra in computer, they use logic gate as fundamental element in circuit. Following table list all the gates available and the corresponding function. Logical Operation Symbol Gate Output NOT ~,-,! A A AND ./^ A A.B B OR +/^ A B NAND (NOT+NOT) A A.B=A.B/A .B B NOR (NOT + OR) A A+B=A.B/A B XOR A AB/AB(A*B) B (A+B). (A+B)=AB+A.B XNOR / A AB/AB B (A+B).(A+B)=A B+A.B Truth table for these gates is given blow: A B + A
Example Simplify 1. (x + y) (x + z) (y + z) 2. xy + xz + yzw 3. A + B +CD 4. ABC + ABC + ABC 5. xy + xy + xy+xyz + xyz Answer (x + y).(x + z) Xy + yz zw + xz (A + B) . (C + D) Connot be simplicied further X + y in example 4 above, all the three variable are present in each of four Products (calling AND operation Product). Such expression in which all the variables (either complemented or otherwise) are connected with AND operation in each term and all terms are connected with OR operator, are said to be expression in sop sum of Product term. Each term of SOP form is called minterm. The dual of SOP is POS (Product of sums) and Dual of minterm is a maxterm. Example: (x,y,z)= xy is not in SOP form because variable z is not present in the product. To convert it is SOP form multiply the term with (z+z)=1 F(x,y,z)=xyz+xy,z Now both terms are minterm and f is in SOP form. IMPLEMENTATIONS OF BOOLEAN ALGERA IN COMPUTER Logic circuits Logic Circuits are used to implement Boolean Algebra in computer, they use logic gate as fundamental element in circuit. Following table list all the gates available and the corresponding function. Logical Operation Symbol Gate Output NOT ~,-,! A A AND ./^ A B A.B OR +/^ A B NAND (NOT+NOT) A B A.B=A.B/A .B NOR (NOT + OR) A B A+B=A.B/A XOR A B AB/AB(A*B) (A+B). (A+B)=AB+A.B XNOR / A B AB/AB (A+B).(A+B)=A B+A.B Truth table for these gates is given blow: A B + A
3.final_01.docx