Properties of logical operators
Other tools in our tool-box
Just like the operations we know - sum, subtraction, multiplication, division) - the logical operators - AND, OR, NOT - have properties too. We are going to learn them so we can use them at a later time.
If you don’t know what these operators are, check out this article:
Duality
If an equation is valid, like A+ 1 = 1, then switching the operator to the AND and the constants (0 → 1 and 1 → 0) will generate a new valid equation.
A + 1 = 1
A · 0 = 0Idempotency
A + A = A
A · A = AInverse
A + !A = 1
A · !A = 0Commutative
A + B = B + A
A · B = B · A Associative
(A + B) + C = A + (B + C) = (A + C) + B
(A · B) · C = A · (B · C) = (A · C) · BDistributive
A + (B · C) = (A · B) + (A · C)
A · (B + C) = (A + B) · (A + C) Absorption
A + A · B = A
A · (A + B) = AA + (!A · B) = A + B
A · (!A + B) = A · B
Proof
A + A · B = A · (1 + B) = A · 1 = A
A · (A + B) = A · A + A · B = A + A · B = A
A + !A · B = (A + !A) · (A + B) = 1 · (A + B) = A + B
A · (!A + B) = A · !A + A · B = 0 + A · B = A · B
De Morgan’s laws
A · B = !(!A + !B)
A + B = !(!A · !B) These tools we have now we’ll let us handle functions better. We will use them to simplify the functions of circuits we are going to design. How? That’s for another time.
Thank you for reading this article! I hope it helped you!
