1. Logic Circuits In today’s lesson we will look at: the symbols for NOT, AND, OR and EOR using truth tables to represent logic circuits two new operators – NAND and NOR using equivalence to simplify circuits the “half-adder” circuit

2. Logic Circuits You might sometimes see diagrams that look like electrical circuits, but which contain symbols for Boolean operators – these are called logic circuits. Each logical operation has its own symbol: NOT

3. Example Circuit Symbols are combined and inputs labelled: Output can be shown using truth tables: D NOT AND OR ABCD 000 001 010 011 100 101 110 111

4. NAND and NOR With logic circuits, there are two new operators that you might come across – NAND and NOR: Their names are short for NOT AND and NOT OR, which reminds us what they do. NAND behaves like AND with a NOT after it, and NOR behaves like an OR with a NOT after it - i.e. the results are the opposite of a normal AND and OR.

5. Truth Table - NOR The NOR operator gives a false result if any of the input values is true, e.g. aba OR b 000 011 101 111 a NOR b 1 0 0 0

6. Truth Table - NAND The NAND operator gives a false result if both of the input values are true, e.g. aba AND b 000 010 100 111 a NAND b 1 1 1 0

7. Why Use NAND and NOR? Remember DeMorgan’s Duals from last week? –NOT(a OR b) = NOT a AND NOT b –NOT(a AND b) = NOT a OR NOT b The left-hand part of those equivalences are NOR and NAND, so they can be used to make logic circuits simpler. Also, NAND is easy to manufacture because a single transistor behaves like a NAND gate, so circuits using NAND can be both logically and physically simpler and more efficient.

8. Simplification Look at this circuit – it uses three components: It represents the operation Y = NOT A OR NOT B However, from the previous slide, we know that: NOT(a AND b) = NOT a OR NOT b We also know that NOT(a AND b) is the same as a NAND b, so the whole circuit can be simplified to a single NAND gate: Y B A

9. Another Example ABCOutput 000 001 010 011 100 101 110 111 AND NAND OR What is the output of this logic circuit?

10. Half-Adder ABDESC 00 01 10 11 What is the output of this circuit? This circuit performs binary addition (C = carry) It also has a simpler equivalent: