Boolean Algebra

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Boolean Algebra
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Logical Statements

• A proposition that may or may not be true
• Today is Monday
• Today is Sunday
• It is raining

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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.

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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.

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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

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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

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Writing AND, OR, NOT

• A AND B A B AB
• A OR B A v B AB
• NOT A A A
• TRUE T 1
• FALSE F 0

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Exercise

• AB AB
• A AND B OR A AND NOT B
• (A B)(B)
• NOT (A OR B) AND B

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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,

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Truth Tables

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

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Truth Table
M R M R MR M R
T T
T F
F T
F F
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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
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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
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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
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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.

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A (A B) AB
A B A B A B A B A(AB) 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
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Exercise

• Write the truth table for (A A) B

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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
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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)

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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

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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
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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

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Identities with NOT

• (A) A
• A A True
• AA False

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DeMorgans Laws

• (A B) AB
• (AB) A B
• Exercise - Simplify the following with identities
• (AB)

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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 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 F T 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.
F T F 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.
F F F 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.
For example, X AB AB For example, X AB AB For example, X AB AB For example, X AB AB
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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 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 F F 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.
F T T 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.
F F F 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.
Solve the Truth Table given above. Solve the Truth Table given above. Solve the Truth Table given above. Solve the Truth Table given above.
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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 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 F F 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.
F T T 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.
F F F 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.
Solution is, X AB AB (A AND B) OR ( NOT A AND B) Solution is, X AB AB (A AND B) OR ( NOT A AND B) Solution is, X AB AB (A AND B) OR ( NOT A AND B) Solution is, X AB AB (A AND B) OR ( NOT A AND B)