De Morgans Theorems Digital Electronics De Morgans Theorems
- Slides: 77
De. Morgan’s Theorems Digital Electronics
De. Morgan’s Theorems are two additional simplification techniques that can be used to simplify Boolean expressions. Again, the simpler the Boolean expression, the simpler the resulting logic. 2
Augustus De. Morgan, an Englishman, born in India in 1806. He was instrumental in the advancement of mathematics and is best known for the logic theorems that bear his name. 3
De. Morgan’s Theorem #1: Proof 0 0 0 1 1 0 1 0 0 1 1 1 0 0 4
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 0 1 1 1 0 0 5
De. Morgan’s Theorem #1 Proof 0 0 1 1 0 0 0 1 1 1 0 0 6
De. Morgan’s Theorem #1 Proof 0 0 1 1 0 0 0 1 1 1 0 0 7
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 8
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 9
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 10
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 11
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 12
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 13
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 1 14
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 0 1 15
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 16
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 17
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 18
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 19
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 20
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0
De. Morgan’s Theorem #1 Proof 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 The truth-tables are equal; therefore, the Boolean equations must be equal!
23
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 24
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 25
De. Morgan's Theorem #2 Proof 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 26
De. Morgan's Theorem #2 Proof 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 27
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 28
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 29
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 30
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 31
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 32
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 33
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 34
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 35
De. Morgan's Theorem #2 Proof 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 1 36
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 37
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 38
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 39
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 40
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 41
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 42
De. Morgan's Theorem #2 Proof 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 The truth-tables are equal; therefore, the Boolean equations must be equal!
Summary Boolean & De. Morgan’s Theorems Commutative Law Associative Law Distributive Law Consensus Theorem De. Morgan’s 44
De. Morgan Shortcut BREAK THE LINE, CHANGE THE SIGN! Break the LINE over the two variables, and change the SIGN directly under the line. For Theorem #14 A, break the line, and change the AND function to an OR function. Be sure to keep the lines over the variables. 45
De. Morgan Shortcut BREAK THE LINE, CHANGE THE SIGN! Break the LINE over the two variables, and change the SIGN directly under the line. For Theorem #14 B, break the line, and change the OR function to an AND function. Be sure to keep the lines over the variables. 46
De. Morgan’s: Example #1 Simplify the following Boolean expression and note the Boolean or De. Morgan’s theorem used at each step. Put the answer in SOP form. 47
De. Morgan’s: Example #1 Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 48
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 49
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 50
De. Morgan’s: Example #1 Theorem #14 A Theorem # & # Theorem #9 Rewritten without AND symbols and parentheses 51
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 52
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 53
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem # Rewritten without AND symbols and parentheses 54
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 55
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 56
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 57
De. Morgan’s: Example #1 Theorem #14 A Theorem #9 & #14 B Theorem #9 Rewritten without AND symbols and parentheses 58
De. Morgan’s: Example #2 Take a look at the VERY poorly designed logic circuit shown below. If you were to analyze this circuit to determine the output function F 2, you would obtain the results shown. 60
De. Morgan’s: Example #2 Simplify the output function F 2. Be sure to note the Boolean or De. Morgan’s theorem used at each step. Put the answer in SOP form. 61
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 62
De. Morgan’s: Example #2 Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 63
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 64
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 65
De. Morgan’s: Example #2 Theorem #14 A Theorem #14 B Theorem #9 Rewritten without parentheses 66
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 67
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 68
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Rewritten without parentheses 69
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 70
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 71
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem # Rewritten without parentheses 72
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 73
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 74
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 75
De. Morgan’s: Example #2 Theorem #14 A Theorem #9 Theorem #14 B Theorem #9 Rewritten without parentheses 76
De. Morgan’s: Example #2 can be simplified to… 77