1. Boolean Algebra

2. Logical Statements A proposition that may or may not be true:  Today is Monday  Today is Sunday  It is raining

3. Compound Statements More complicated expressions can be built from simpler ones:  Today is Monday AND it is raining.  Today is Sunday OR it is NOT raining  Today is Monday OR today is NOT Monday (This is a tautology)  Today is Monday AND today is NOT Monday (This is a contradiction) The expression as a whole is either true or false.

4. Things can get a little tricky… Are these two statements equivalent?  It is not nighttime and it is Monday OR it is raining and it is Monday.  It is not nighttime or it is raining and Monday AND it is Monday.

5. Boolean Algebra Boolean Algebra allows us to formalize this sort of reasoning. Boolean variables may take one of only two possible values: TRUE or FALSE.  (or, equivalently, 1 or 0) Algebraic operators: + - * / Logical operators - AND, OR, NOT, XOR

6. Logical Operators A AND B is True when both A and B are true. A OR B is always True unless both A and B are false. NOT A changes the value from True to False or False to True. XOR = either a or b but not both

7. Writing AND, OR, NOT A AND B = A ^ B = AB A OR B = A v B = A+B NOT A = ~A = A’ TRUE = T = 1 FALSE = F = 0

8. Exercise AB + AB’ A AND B OR A AND NOT B (A + B)’(B) NOT (A OR B) AND B

9. Boolean Algebra The = in Boolean Algebra means equivalent Two statements are equivalent if they have the same truth table. (More in a second) For example,  True = True,  A = A,

10. Truth Tables Provide an exhaustive approach to describing when some statement is true (or false)

11. Truth Table MR M’R’MRM + R TT TF FT FF

12. Truth Table MRM’R’MRM + R TTFF TFFT FTTF FFTT

13. Truth Table MRM’R’MRM + R TTFFT TFFTF FTTFF FFTTF

14. Truth Table MRM’R’MRM +R TTFFTT TFFTFT FTTFFT FFTTFF

15. Example Write the truth table for A(A’ + B) + AB’ Fill in the following columns:  A, B, A’, B’, A’ + B, AB’, A (A’ + B), whole expression.

16. A (A’ + B) + AB’ AB A’ B’ A’ + BA B’A(A’+B)Whole TTFFTFTT TFFTFTFT FTTFTFFF FFTTTFFF

17. Exercise Write the truth table for (A + A’) B

18. Solution to (A + A’) B ABA’A + A’ (A + A’) B TTFTT TFFTF FTTTT FFTTF

19. Boolean Algebra - Identities A + True = True A + False = A A + A = A A + B = B + A (commutative) A AND True = A A AND False = False A AND A = A AB = BA (commutative)

20. Associative and Distributive Identities A(BC) = (AB)C A + (B + C) = (A + B) + C A (B + C) = (AB)+(AC) A + (BC) = (A + B) (A + C) Exercise: using truth tables prove -  A(A + B) = A

21. Solution: A AND (A OR B) = A ABA + BA (A + B) TTTT TFTT FTTF FFFF

22. Using Identities A + (BC) = (A + B)(A + C) A(B + C) = (AB) +(AC) A(A + B) = A A + A = A Exercise - using identities prove:  A + (AB) = A  A +(AB) = (A +A)(A + B)  = A (A + B) = A

23. Identities with NOT (A’)’ = A A + A’ = True AA’ = False

24. DeMorgan’s Laws (A + B)’ = A’B’ (AB)’ = A’ + B’ Exercise - Simplify the following with identities  (A’B)’

25. Solving a Truth Table ABXWhen you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR. TTT TFT FTF FFF For example, X = AB + AB’

26. Exercise: Solving a Truth Table ABXWhen you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR. TTT TFF FTT FFF Solve the Truth Table given above.

27. Exercise: Solving a Truth Table ABXWhen you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR. TTT TFF FTT FFF Solution is, X = AB + A’B = (A AND B) OR ( NOT A AND B)