3 further logic concepts
- 1. 1 Further logic concepts • NAND, NOR, XOR gates, utility of universal logic functions,Boolean expressions of NAND and NOR gate • How to derive output Boolean expression of given logic circuit. • How to construct truth table given Boolean expression • How to draw logic circuit schematic given Boolean expression. • Obtain output Boolean Expressions in SOP and POS form
- 2. 2 Variable, Literal (in Boolean Expressions) • Variable is a symbol used to represent a logical quantity • Any variable can have a 1 or 0 value • The complement of a variable A is indicated by Ā or A’ (read as A bar) • Literal is a variable or the complement of a variable
- 3. 3 Boolean Expressions of Basic Gates Input A B Output X= A+B 0 0 0 1 1 0 1 1 0 1 1 1 Truth table for OR gate Input A B Output X=A.B 0 0 0 1 1 0 1 1 0 0 0 1 Truth table for AND gate Input A Output X=Ā Can be written as X= A’ 0 1 1 0 Truth table for NOT gate
- 4. 4 Boolean Addition • Boolean Addition is equivalent to the OR operation • Basic rules for Boolean Addition: 1 + 1 = 1 1 + 0 = 1 0 + 1 = 1 0 + 0 = 0 • A sum term is equal to 1 when one or more literals in the term are 1 • A sum term is equal to 0 only if each of the literals is 0
- 5. 5 Boolean Multiplication • Boolean Multiplication is equivalent to the AND operation • Basic rules for Boolean Addition: 1.1 = 1 1.0 = 0 0.1 = 0 0.0 = 0 • A product term is equal to 1 only if each of the literals in the term is 1 • A product term is equal to 0 when one or more literals are 0
- 6. 6 NAND Gate, NOR Gate Input A B Output Y=(A.B)’ 0 0 0 1 1 0 1 1 1 1 1 0 Truth table for NAND gate Input A B Output Y=(A+B)’ 0 0 0 1 1 0 1 1 1 0 0 0 Truth table for NOR gate
- 7. 7 Boolean Expression of a Logic Circuit • To derive the Boolean Expression for a given logic circuit, begin at the left-most inputs and work towards the final output by writing the expression for each gate A B C Y= A.B C A.B.C A.B.C + A
- 8. 8 Constructing Logic Circuit Given Boolean Expression • Identify the logic gates that you need from the Boolean Expressions • Connect the gates in such a manner to obtain the desired output in the given Boolean Expression CBCACABy ++=The given equation is:
- 9. 9 Constructing Logic Circuit Given Boolean Expression (Example) • Draw the logic circuit schematic based on the below Boolean expression. ).).(( ACBBAy ++= ` A B C y ).).(( ACBBAy ++= )( BA+ ).( ACB + A B A CB.( B C
- 10. Sum-of-Products (SOP) • Two or more product terms summed by Boolean Addition • A single bar cannot extend over more than one variable DBACBAX .... +=DBACBAX .... += ✓ • Implementation requires OR to combine the outputs of two or more AND gates
- 11. Product-of-Sum (POS) • Two or more sum terms multiplied • A single bar cannot extend over more than one variable ✓ • Implementation requires AND to combine the outputs of two or more OR gates )).(( DBACBAX ++++=)).(( DBACBAX ++++=
- 12. Standard SOP from Truth Table • List the binary values of the input variables for which the output is 1 • Convert each binary value to the corresponding product term by: ✓ Replacing 1 with corresponding variable ✓ Replacing 0 with corresponding variable complement • These product terms which are composed of every input variable or its complement ANDed together are known as minterms • Sum these minterms together
- 13. Standard POS from Truth Table • List the binary values of the input variables for which the output is 0 • Convert each binary value to the corresponding sum term by: ✓ Replacing 0 with corresponding variable ✓ Replacing 1 with corresponding variable complement • These sum terms which are composed of every input variable or its complement ORed together are known as maxterms • Multiply these maxterms together
- 14. 14 Constructing Truth Table from Boolean Expressions (Example) • Evaluate the Boolean expression for all possible combinations of values for the input variables Y = 1 when A=0, C=0, D=0, B=X (does not matter/don’t care) OR when B=1, C=1, D=0, A=X (does not matter/don’t care) OR when A=1, B=1, C=0, D=X (does not matter/don’t care) A B C D Y 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 A B C D Y DCA DBC CAB CABDBCDCAY ++=