Boolean Algebra Monday/Wednesday 7th Week

Logical Statements Today is Friday AND it is sunny. Today is Friday AND it is rainy. Today is Monday OR it is sunny. Today is Monday OR it is raining. Today is Friday OR it is NOT raining.

A More Challenging Example Are these two statements the same? It is NOT Friday OR it is raining. It is NOT the case that it is Friday AND it is NOT raining.

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 Algebraic operators: + - * / Logical operators - AND, OR, NOT, XOR, NOR, NAND

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 NOR = NOT OR NAND = NOT AND

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

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

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

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

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

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

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

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

Example Write the truth table for A(A’ + B) + AB’ (p 266, exercise #3a) First, write in words: A AND (NOT A OR B) OR (A AND NOT B) Then do a truth table with the following columns: A, B, A’, B’, A’ + B, AB’, A (A’ + B), whole expression.

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

Exercise Write the truth table for (A + A’) B First, write in words. Then do a truth table.

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

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

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

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

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

Identities with NOT (A’)’ = A A + A’ = True AA’ = False On and on and on and on …

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

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’ = (A AND B) OR ( A AND NOT B)

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.

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)