Boolean Algebra

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Boolean Algebra

• Computer Science 1611

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AND

• (today is Monday) AND (it is raining)
• (today is Monday) AND (it is not raining)
• (today is Friday) AND (it is raining)
• (today is Friday) AND (it is not raining)

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OR

• (today is Monday) OR (it is raining)
• (today is Monday) OR (it is not raining)
• (today is Friday) OR (it is raining)
• (today is Friday) OR (it is not raining)

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NOT

• (today is Monday) OR NOT (it is raining)
• (today is Monday) OR (it is raining)
• (today is Friday) AND NOT(it is raining)
• (today is Friday) AND (it is raining)
• (today is Monday) AND NOT(it is raining)
• (today is Monday) AND (it is raining)
• (today is Friday) OR NOT (it is raining)
• (today is Friday) OR (it is raining)

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IMPLIES (A ? B)

• A ?B is False only when A is True and B is False.
• In other words, A ? B is True except when the
  premise (A) is True and the conclusion (B) is
  False.
• A ? B is logically equivalent to
• (NOT A) OR B

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Truth Table (AND, OR, NOT)
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Truth Table (AND, OR, NOT)
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Truth Table (AND, OR, NOT)
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Truth Table (AND, OR, NOT)
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Truth Tables

• A AND B is True only 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.

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IMPLIES (A ? B)
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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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Example

• Write the truth table for A(A B) AB
  (section 7.5, AE, p 308, 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, NOT A, NOT B, NOT A OR B, A AND NOT B, A
  AND (NOT A OR B), whole expression.

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X A (A B) AB
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Exercise

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

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Solution to (A A) B
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Boolean Algebra

• Boolean Algebra is made up of two constants (True
  and False)
• Several operators - AND, OR, NOT, XOR, NOR, NAND
• XOR either a or b but not both
• NOR NOT OR
• NAND NOT AND

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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,

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Boolean Algebra - Identities

• A OR True True
• A OR False A
• A OR A A
• A OR B B OR A
• (commutative)

• A AND True A
• A AND False False
• A AND A A
• A AND B B AND A
• (commutative)

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Associative and Distributive Identities

• A AND (B AND C) (A AND B) AND C
• A OR (B OR C) (A OR B) OR C
• A OR (B AND C) (A OR B) AND (A OR C)
• A AND (B OR C) (A AND B) OR (A AND C)
• Exercise using truth tables prove -
• A AND (A OR B) A

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Solution A AND (A OR B) A

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Using Identities

• A OR (B AND C) (A OR B) AND (A OR C)
• A AND (B OR C) (A AND B) OR (A AND C)
• A AND (A OR B) A
• A OR A A
• Exercise - using identities prove
• A OR (A AND B) A
• A AND (A OR B) A
• A OR (A AND B) (A OR A) AND (A OR B)

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Identities with NOT

• NOT (NOT A) A
• A OR NOT A True
• A AND NOT A False
• On and on and on and on

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DeMorgans Laws

• NOT (A OR B) NOT A AND NOT B
• (A B) (A) (B)
• NOT (A AND B) NOT A OR NOT B
• (AB) A B
• Exercise - Simplify the following with identities
• NOT (NOT A AND B)

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Solving a Truth Table
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Solving a Truth Table
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Exercise 1 Solving a Truth Table
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Exercise1 Solving a Truth Table
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Exercise2 Solving a Truth Table