Boolean Algebra Logical Statements n A proposition that

Boolean Algebra

Logical Statements n A proposition that may or may not be true: Today is Monday ¨ Today is Sunday ¨ It is raining ¨

Compound Statements n 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 ¨ n ¨ Today is Monday AND today is NOT Monday n n (This is a tautology) (This is a contradiction) The expression as a whole is either true or false.

Things can get a little tricky… n 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. ¨

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

Logical Operators n n 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

Writing AND, OR, NOT n n n 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 n n AB + AB’ A AND B OR A AND NOT B (A + B)’(B) NOT (A OR B) AND B

Boolean Algebra n n n 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, ¨

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

Truth Table M R T T T F F M’ R’ MR M+R

Truth Table M R M’ R’ T T F F T F T T F F F T T MR M+R

Truth Table M R M’ R’ MR T T F F T T F F T T F M+R

Truth Table M R M’ R’ MR M +R T T F F T F T T F F

Example n n 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.

A (A’ + B) + AB’ A B A’ B’ A’ + B A B’ A(A’+B) Whole T T F F T F T F T T F F F T T T F F F

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

Solution to (A + A’) B A’ A + A’ (A + A’) B T T F T T T F F F T T T F

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

Associative and Distributive Identities n n n 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

Solution: A AND (A OR B) = A A B A+B A (A + B) T T T F F F

Using Identities n n n 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 n n n (A’)’ = A A + A’ = True AA’ = False

De. Morgan’s Laws n n n (A + B)’ = A’B’ (AB)’ = A’ + B’ Exercise - Simplify the following with identities ¨ (A’B)’

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

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

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