View
220
Download
0
Category
Preview:
Citation preview
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
M R M’ R’ MR M + R
T T
T F
F T
F F
Truth Table
M R M’ R’ MR M + R
T T F F
T F F T
F T T F
F F T T
Truth Table
M R M’ R’ MR M + R
T T F F T
T F F T F
F T T F F
F F T T F
Truth Table
M R M’ R’ MR M +R
T T F F T T
T F F T F T
F T T F F T
F F T T F F
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’
A B A’ B’ A’ + B A B’ A(A’+B) Whole
T T F F T F T T
T F F T F T F T
F T T F T F F F
F F T T T F F F
Exercise
Write the truth table for (A + A’) B First, write in words. Then do a truth table.
Solution to (A + A’) B
A B A’ A + A’ (A + A’) B
T T F T T
T F F T F
F T T T T
F F T T F
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
A B A + B A (A + B)
T T T T
T F T T
F T T F
F F F F
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
A B X When 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.
T T T
T F T
F T F
F F F
For example,
X = AB + AB’ = (A AND B) OR ( A AND NOT B)
Exercise: Solving a Truth Table
A B X When 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.
T T T
T F F
F T T
F F F
Solve the Truth Table given above.
Exercise: Solving a Truth Table
A B X When 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.
T T T
T F F
F T T
F F F
Solution is,
X = AB + A’B = (A AND B) OR ( NOT A AND B)
Recommended