XOR Operator

1. XOR XOR Operator • A short digression… • … to introduce another Boolean operation: exclusive-OR (XOR)

2. XOR Operator • Also referred to as an “odd” function since it returns a 1 only when an odd number of 1’s are input

3. Simplification • Using the axioms to “prove” that a simplified version of a circuit is equivalent to the complex version takes a special kind of person… • …of which I’m not one • Fortunately, there’s another way…

4. Karnaugh Maps • Also known as K-Map • Recall that an expression can be written in the form F(A,B,C) = Σ(0,2,4,5,6) • Which means the functional value is 1 at binary input patterns 0, 2, 4, 5, 6 and 0 at all other input patterns • What does the truth table look like?

5. K-Maps • F(A,B,C) = Σ(0,2,4,5,6) is called a “sum of minterms” representation • The expression for such a representation is F(A,B,C) = A’B’C’ + A’BC’ + AB’C’ + AB’C + ABC’ • We could simplify this via the axioms, right? (assuming we were that special kind of person) • It’s painful!!!

6. BC B A A A 0 1 0 1 00 01 11 10 0 1 0 0 1 0 1 3 2 0 1 3 2 4 5 7 8 1 CD AB 00 01 11 10 0 1 3 2 00 01 4 5 7 8 11 12 13 15 14 10 8 9 11 10 K-Maps • A K-Map is a grid (map) where each square corresponds to a minterm Note the ordering here is Gray code, not binary

7. K-Maps • Notice how neighboring squares (minterms) differ by a single bit…this is the key to the whole thing • Consider minterms 1 and 3 • 1: A’B’C • 3: A’BC • If we were to OR these together • (A’B’C + A’BC) would simplify to A’C via the axioms

8. K-Maps • Great, now what do we do with them? • Place 1’s on the squares that correspond to minterms in the truth table • Place 0’s on all other squares • Group adjacent 1’s into the largest group whose size is a power of 2

9. K-Maps • Notes: • Adjacencies wrap top-to-bottom and left-to-right • 1’s can be part of more than one group • When you are grouping adjacent squares you’re essentially applying axiom 4 (x + x’ = 1) so the variable that is being “spanned” can be removed from the minterm

10. Simplification via Axioms(aka Proofs) • Here’s a little insight that no one ever taught me F(x, y, z) = xy’z + x’y’z + x’yz • Notice how the middle term shares two elements with each of the others • Using association, distribution, and inverse: F(x, y, z) = y’z + x’yz • One more application of distribution F(x, y, z) = z(y’ + x’y) • We could have arrived at a similar solution by grouping the 2nd two terms

11. yz x 00 01 11 10 0 0 1 1 0 1 0 1 0 0 Simplification via Axioms(aka Proofs) • But, can we do better? • Notice that we use the minterm x’y’z in two groupings • What does that mean in terms of an axiomatic proof? F(x, y, z) = y’z + x’z = z(x’ + y’)

12. Simplification via Axioms(aka Proofs) • It means exactly this… F(x, y, z) = xy’z + x’y’z + x’yz F(x, y, z) = xy’z + x’y’z + x’y’z + x’yz • …by idempotence over OR • Now we can form two associative groupings and arrive at the same answer that the Karnaugh Map gave us

13. What is the truth-table? What is the expression in sum-of-minterms form? What is the simplified expression? What is the (schematic) logic gate implementation? CD AB 00 01 11 10 00 1 0 0 1 01 0 0 0 0 11 0 0 0 0 10 1 0 0 1 Karnaugh Maps

14. Sum-of-Products • This is what we previously called the “sum-of-minterms” • Form the largest power-of-two groupings of 1’s on the K-map • Create the schematic

15. Product-Of-Sums • Instead of forming large adjacent groups of 1’s (on the K-map), form large adjacent groups of 0’s • What does this mean in terms of the original expression/truth-table? • It means you have simplified F’, instead of F • To “fix” what you’ve done you need only negate the final result them apply De Morgan’s theorem

16. Example – Sum-of-Products • F(A,B,C,D) = Σ(0,1,2,5,8,9,10) • Form the truth-table • Form the K-map • Simplify the K-map using sum-of-products • Formulate the boolean expression • Draw the schematic diagram

17. Example – Sum-of-Products B’ D’ F C’ A’ D

18. Example – Product-of-Sums • F(A,B,C,D) = Σ(0,1,2,5,8,9,10) • Form the truth-table • Form the K-map • Simplify the K-map using product-of-sums • Formulate the boolean expression • Negate, apply De Morgan’s • Draw the schematic diagram

19. Example – Product-of-Sums B’ D A’ F C’ D’

20. So What? • As it turns out, the sum-of-products can be easily implemented with NAND gates • Similarly, the product-of-sums can be easily implemented with NOR gates • This may greatly simplify the design thus saving us money!

21. B’ B’ D D’ A’ C’ C’ A’ D’ D NAND/NOR Implementations

22. Combinational Circuits • Definition: A connected arrangement of logic gates with a set of inputs and outputs • Specifically, they have no memory! • Basically, it’s the stuff we’ve been working on so far

23. Augend Sum Half-Adder Addend Carry Combinational Circuit Design • Design a Half-Adder • A combinational circuit that adds 2 bits • Input 1 is call the “Augend” • Input 2 is called the “Addend” • Output 1 is called the “Sum” • Output 2 is called the “Carry”

24. Carry-in Sum Augend Full-Adder Addend Carry-out Combinational Circuit Design • Design a Full-Adder • A combinational circuit that adds 3 bits • Input 1 is call the “Augend” • Input 2 is called the “Addend” • Input 3 is call the “Carry-in” • Output 1 is called the “Sum” • Output 2 is called the “Carry-out”

25. Homework • Pages 37, 38: 1-8, 1-9, 1-10, 1-12, 1-13 • Due Thursday (next lecture)