1.4.3 Boolean Algebra
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What are the AND (∧) identities?
A ∧ 0 ⇒ 0 A ∧ 1 ⇒ A
What are the OR (∨) identities?
A ∨ 0 ⇒ A A ∨ 1 ⇒ 1
What are the negation identities?
A ∧ ￢A ⇒ 0 A ∨ ￢A ⇒ 1
what is the absorption law?
A ∧ (A ∨ B) ⇒ A A ∨ (A ∧ B) ⇒ A
What is the distribution law?
A ∧ (B ∨ C) ⇒ (A ∧ B) ∨ (A ∧ C) (like multiplying out brackets)
what are De Morgan’s laws?
￢(A ∨ B) ⇒ ￢A ∧ ￢B ￢(A ∧ B) ⇒ ￢A ∨ ￢B
What is a half adder?
A circuit which adds two single bit binary numbers together
what are the input and outputs of a half adder?
Inputs: - two single bit binary numbers (eg, A and B) Outputs: - S - the sum of the two numbers - C - the carry bit
what is the logic diagram for a half adder composed of?
S = A XOR B C = A AND B
How can you tell if a circuit is a half adder?
it has 2 inputs
what is a full adder?
a circuit used to add larger binary numbers together.
What are the inputs and outputs of a full adder?
Inputs: - two single bit binary numbers (eg, A and B) - the carry input bit (Cin) Outputs: - the Sum - S - the carry output bit (Cout)
how can you tell a circuit is a full adder?
it has 3 inputs
what is a full adder made up of?
two half adders connected by an OR gate
what is a D-Type flip flop?
A circuit which stores one bit of data
what are the inputs and outputs of a D-type flip flop?
• data input • clock input • output of Q and NOT Q • Q is set to the data input on the rising edge of the clock and stays that value for one clock cycle
How do you use a Karnaugh map to simplify Boolean expressions?
• split expressions by the ORs and add 1s where each section is true • draw boxes that are powers of 2 (eg, 1, 2, 4 etc 1s in them) • look at each box and see which variables stay the same • put each expression from each box together with ORs
what is important to remember for Boolean simplification?
AND takes precedence over OR so: A ∧ B ∨ C ⇒ (A ∧ B) ∨ C