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# Minterms

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• 1. LOGIC DESIGN PART 3 Minterm
• 2. • Larger logic problems require a systematic approach for solution. Modern integrated circuit chips can use millions of logic devices. The sheer magnitude of these designs is a clear sign that a formal approach to the design is needed. • Given a Boolean function described by a truth table, – determine the smallest sum of products function that has the same truth table. – determine the AND-OR-NOT circuit that implements that smallest sum of products function.
• 3. Minterm Expansions • Different ways of expressing a Boolean function can have widely varying levels of complexity. • More complex circuits will require more gates and inverters, so it's a reasonable goal to learn how to devise circuits that are as simple as possible. • In this section we are going to look at how you can represent circuits differently using
• 4. • Let's look at a simple Boolean function of three variables. We'll describe this function with a truth table. The input variables are X, Y and Z, and the function output is F. Let's examine this function in some detail. The only non-zero entries are at: X = 0, Y = 1, Z = 0 and X = 1, Y = 0, Z = 1 The function is 1 for those two input conditions and zero for all other input conditions.
• 5. How we can implement this function • We want the output to be 1 whenever we have either – X=0 AND Y=1 AND Z=0 • OR when we have – X=1 AND Y=0 AND Z=1. • This word statement is very close to the function we want. We've highlighted the important aspects of the function. Here's the function: • This function is read as (NOT-X AND Y AND
• 6. Defining Minterms • This form is composed of two groups of three. Each group of three is a minterm. Important points about minterms include the following. – In a minterm, each variable, X, Y or Z appears once, either as the variable itself or as the inverse. – Each minterm corresponds to exactly one entry (row!) in the truth table.
• 7. • To build any Boolean function from minterms do the following. – Get a truth table for the function – For each entry of the truth table for which the function takes on a value of 1, determine the corresponding minterm expression remembering that every variable of its inverse will appear in every minterm. – OR all the minterms from the second step together.
• 8. IN SHORT ! • A truth table gives a unique sum-of-products function that follows directly from expanding the ones in the truth table as minterms.
• 9. An example using Minterms • Three young graduates have formed a company. The three graduates, Alisha, Ben and Corey have a system to minimize friction. For all minor decisions they want to use a circuit that will determine when a majority of the three of them has voted for a proposal. Essentially, they want a box with three inputs that will produce a 1 at the output
• 10. STEP 1 Get the truth table T1 T2 T3 T4
• 11. STEP 2 • Identify the minterms
• 12. STEP 3 ORing all together
• 13. • Now we need to simplify our equation further more. • For this we require basic Boolean algebra first
• 14. Simplifying the Circuit FIRST POSSIBILITY
• 15. SECOND POSSIBILITY If you OR anything with itself, you get the original quantity back (X + X = X).
• 16. • Here is the three-variable truth table and the corresponding minterms:
• 17. Example • Example: suppose a function F is defined by the following truth table Since F= 1 on rows 1, 2, 4, and 7, we obtain A compact notation is to write only the numbers of the minterms included in , using the Greek letter capital sigma to indicate a sum: