(5 pts) Exercise B-1 • Show the truth table for NAND and NOR gates NOR NAND A 0 0 1 1 B 0 1 0 1 x 1 1 1 0 A 0 0 1 1 B 0 1 0 1 x 1 0 0 0 (5 pts) Exercise B-2 • A.) Show the truth table for the following logic circuit A A C y 000 0 001 0 010 1 011 0 100 B.) Write the Boolean equation for this circuit. 1 101 0 110 0 111 0 B y C • B y = (A xor B) C (5 pts) Exercise B-3 • Draw a circuit for the following formula: F = ( (A + B) C ) + D A B C D (2 pts EXTRA CREDIT) Exercise B-4 • • Recall – how many entries are in a truth table for a function with n inputs? Consider – how many different truth tables are possible for a function with n inputs? The truth table has 2^n entries. For each entry, I can choose a 0 or a 1. Thus there are 2^n choices to make, and each choice has 2 options, so total number of possibilities is: 2^(2^n) (5 pts) Exercise B-11 • Show the sum of products for the following truth table. A B C f 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 z ( A B C) ( A B C) ( A B C) ( A B C) (5 pts) Exercise B-12 • Simplify the following equations (use Boolean laws discussed earlier) B( A 0) AB B( AA ) 0 ( A B )( A B) AB A B ( A B) ( A B C ) ABC Is AB No the same as AB ? (5 pts) Exercise B-13 • Use bubble pushing to simplify this circuit (10 pts) Exercise B-14 • A) Show the sum of products for the following truth table. A B C f 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 z ( A B C) ( A B C) ( A B C) ( A B C) ( A B C) ( A B C) • B) Simplify this equation z ( A B C) ( A B C) A z (B C) (B C) A (8 pts) Exercise B-15 • Simplify the following equations C ( A 1) C(1) = C AB( A C ) ABA + ABC = AB + ABC = AB(1+C) = AB ( A B )( A C ) AA A B AC B C A B AC B C ( B 0)(C D 1) (B)(1) = B (5 pts) Exercise B-21 • • 1. Fill in the following K-Map based on the truth table at right 2. Minimize the function using the K-map A A BC 1 1 f AB AC BC 0 1 A B C f 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 BC 0 0 BC 1 0 (5 pts) Exercise B-22 • Draw the two-level circuit for the function from Exercise B-21 A B C (5 pts) Exercise B-23 • Suppose we already have this k-Map. Minimize the function. CD AB 1 AB 1 AB 1 AB 0 C D CD CD 0 0 1 1 1 1 1 0 0 0 0 1 f AB AD BC BC D (3 pts) Exercise B-24 • Consider your answer to Exercise B-23. Using a K-map, you found some particular two-level, minimal circuit. a.) Is your answer unique? In other words, is there only one possible twolevel circuit for that K-Map that is minimal, or is there another one that is logically different but still correct and just as small? Circle one: UNIQUE NOT UNIQUE b.) Will this always be the case, or could a different K-map change your answer? If you use a K-map and reduce as much as possible, then this will find a two-level circuit of minimal size. However, is this “unique” – or is there more than one possible solution? a.) For the function above, only one way to minimize, so YES, it is unique. b.) NO, this is not always true. For example, look at the solution to B-25. There are three different solutions that are equally minimal, but different. (10 pts) Exercise B-25 • Suppose we already have this k-Map. Minimize the function. See note below on how to group this one CD AB 1 AB 0 AB 1 AB 1 AC D C D CD CD 0 1 1 AC 1 1 1 1 1 0 0 1 0 BD CD Note: one more grouping is not shown – the “1” at top left must combine with either the one at top right OR the one at bottom left – that’s why there are two answers. OR f AC D BD CD AC AB D f AC D BD CD AC BC D OR a third way – handling the “groups of two” differently from how circled above: f ABC BD CD AC BC D