Boolean Algebra Flashcards
¬(¬A) =
A
A∧ ¬A =
0
A∨ ¬A =
1
conjunction
AND
disjunction
OR
Exclusive disjunction
XOR
Negation
NOT
∧
AND (*)
∨
OR (+)
De Morgan’s Law (first)
¬ (A∨B) = (¬ A) ∧(¬ B)
NOT (A OR B) is the same as (NOT A) AND (NOT B)
Distribution LAW (OR)
A∧(B∨C)=(A∧B)∨(A∧C) A AND (B OR C) is the same as (A AND B) OR (A AND C)
Distribution LAW (AND)
A∨(B∧C)= (A∨B)∧(A∨C) A OR (B AND C) is the same as (A OR B) AND (A OR C)
Association LAW (OR)
A∨(B∨C)=(A∨B)∨C= A∨B∨C A OR (B OR C) is the same as (A OR B) OR C is the same as A OR B OR C
Association LAW (AND
A∧(B∧C)=(A∧B)∧C= A∧B∧C A AND (B AND C) is the same as (A AND B) AND C is the same as A AND B AND C
commutation LAW AND
A∧B = B∧A
The order in which two variables are AND’ed makes no difference
commutation LAW OR
A∨B = B∨A
The order in which two variables are OR’ed makes no difference
Absorption Law AND
A∨(A∧B) = A X OR (X AND Y) is the same as X
Absorption Law OR
A∧(A∨B)=A X AND (X OR Y) is the same as X
X∧0=
0
X∧1=
X
X∧X=
X
X∧¬X=
0
X∨0 =
X
X∨1 =
1
X∨X=
X
X∨¬X=
1
AND
The output it true if both inputs are true, else its false
OR
The output is true if 1 or both inputs are true, else the output is false
NOT
reverses the input
XOR
The output is true if one input is true, else the output is false
What does binary represent?
the presence or absence of current
logic gate
a series of electrical switches that take one or more inputs and produce a single output
¬
NOT
⊕ ⊻
XOR
De Morgan’s Law (second)
¬ (A∧B) = (¬ A)∨(¬ B)
NOT (A AND B) is the same as (NOT A) OR (NOT B)
Karnaugh map rules
- no zeros in the blocks
- no diagonal blocks
- groups as large as possible
- groups contain 2^n blocks
- overlapping blocks are allowed
- wrap around blocks are allowed
- aim for the smallest number of groups
Half adder use
adds two bits
Half adder outputs/inputs
inputs: the two bits to be added
outputs: the result and the carry
logic gates in a half adder
XOR (to the sum)
AND (to the carry)
limitations of half adders
- can only add one bit numbers
- only two inputs, the carry from a previous addition can’t be incorperated
Full adder
combines two half adders, used to add a series of bits
Full adder inputs/outputs
inputs: the two bits to add, the previous carry bit
outputs: the result, the carry
Flip flop
elemental sequential logic circuit that can store one bit and flip between two states
Flip Flop use
used as a memory cell to store the state of a bit
clock
sequential circuit that changes state at regular time intervals
clock use
synchronise the change of state of flip flop circuits
flip flop clock diagrams
on each rising edge of the clock change the result to whatever the input is (it may not need changing)
example uses of flip flops
- in registers
- static RAM
flip flop logic gates
Two NAND gates with two inputs. The result of of each is the second input for the other.
D-type flip flop
two flip flops combined
how is a full adder different to a half adder?
a full adder is two half adders which is able to add three bits: the two inputs from the first half adder and the carry.
