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

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Presentation on theme: "Boolean Algebra. Logical Statements A proposition that may or may not be true:  Today is Monday  Today is Sunday  It is raining."— Presentation transcript:

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)


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