Boolean Algebra
VLSI → Combinational Logic
PrepFusion · Digital Electronics · Marathon 1
Start with the suggested sections below. The full lecture is 11h 54m; each section opens at its topic and stops at the end of that section. Neso Academy is available for another explanation.
Watch this section on YouTube ↗Flip a bit. Watch the whole circuit respond.
AND needs every input HIGH. OR needs any input HIGH. NOT inverts its input.
Notation & precedence
+ means OR, · means AND, and ′ means NOT. Apply NOT, then AND, then OR. In X + YZ, the AND gate evaluates YZ first.
Test a law instead of memorizing a table
Choose an identity, then change the inputs. Both sides must agree for every combination.
X + XY1X1The extra branch adds no new HIGH cases.
What is a dual?
Swap AND with OR and 0 with 1 throughout an identity. Variables and their complements keep their names. The transformed identity is also valid.
Move the inversion. Keep the output.
Invert both inputs and swap the gate. Toggle X and Y to compare the two networks.
(XY)′
X′ + Y′
Same truth function · different gate arrangement
Build an expression by painting the outputs
Tap F to change the function. SOP collects the HIGH rows; POS collects the LOW rows.
X′Y + XY↓ simplifies toF = YWhy do POS literals look reversed?
A maxterm must be 0 on its selected row. For X=0, Y=1, the term (X + Y′) is 0. In SOP, the minterm X′Y is 1 on that same row.
Shrink three product terms to one wire
Step through the rewrite. Change X, Y, or Z at any point: the output still follows Y.
XYZ′ + X′YZ′ + YZStart with three product terms.
Verify every input
Your turn: remove a redundant branch
What does X + XY simplify to?
Completion saves automatically after you read through to the end.
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