Introduction to Multilingual Retrieval Augmented Generation (RAG)
IS 151 lecture 4
1. Integrated Circuit (IC) Logic
Families
• Integrated – composed of various components
• 3 digital IC families
– TTL – Transistor-Transistor Logic
• The use of bipolar junction transistors in the circuit
technology used to construct the gates at the chip level.
– CMOS – Complementary Metal Oxide Semiconductor
• Uses field effect transistors; logic functions are the same
(whether the device is implemented with TTL or CMOS), the
difference comes in performance characteristics.
– ECL – Emitter Coupled Logic
• Bipolar circuit technology; has the fastest switching speed
but it’s power consumption is much higher
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2. Boolean Operations and
Expressions
• Boolean algebra – the mathematics of digital
systems
• Variable – a symbol used to represent a logical
quantity.
• Any single variable can have a 0 or a 1 value.
E.g. A, B, C
• Complement – the inverse of a variable
– E.g. A’, B’, C’
– If A = 0, A’ = 1 and vice versa
• Literal – a variable or the complement of a
variable
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3. Boolean Addition
• Equivalent to the OR operation
• Basic rules
•
•
•
•
0+0=0
0+1=1
1+0=1
1+1=1
• e.g. determine the values of A, B, C and D which makes
the sum term A + B’ + C + D’ equal to 0.
• Solution: for the sum to be 0, each of the literals on the term
must be 0. Therefore A = 0, B = 1 (so that B’ = 0), C = 1, D =
1
• Exercise: determine the values of A and B which makes
the sum term A’ + B = 0 (A = 1, B = 0)
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4. Boolean Multiplication
• Equivalent to the AND operation
• Basic rules
•
•
•
•
0.0 = 0
0.1 = 0
1.0 = 0
1.1 = 1
• e.g. determine the values of A, B, C and D which make
the product A.B’.C.D’ equal to 1
• Solution: for the product term to be 1, each one of the literals
in the term must be 1. Therefore A = 1, B = 0, C = 1, D = 0
• Exercise: determine the values of A and B which make
the product A’B’ equal to 1 (A = 0, B = 0)
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5. Laws and Rules of Boolean
Algebra
• Laws
– Commutative Laws
• A + B = B + A; AB = BA
– Associative Laws
• A + (B + C) = (A + B) + C; A(BC) = (AB)C
– Distributive Laws
• A(B +C) = AB + AC
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6. Laws and Rules of Boolean
Algebra
•
Rules
1. A + 0 = A
2. A + 1 = 1
3. A.0 = 0
4. A.1 = A
5. A + A = A
6. A + A’ = 1
7. A.A = A
8. A.A’ = 0
9. A’’ = A
10. A + AB = A
11. A + A’B = A + B (same as A + A’B’ = A + B’)
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7. De Morgan’s Theorems
• The complement of a product of variables is
equal to the sum of the complements of the
variables: (XY)’ = X’ + Y’
• The complement of a sum of variables is equal
to the product of the complements of the
variables: (X +Y)’ = X’.Y’
• Example: Apply De Morgan’s theorems to the
expressions
– (XYV)’ = X’ + Y’ + Z’
– (X + Y + Z)’ = X’.Y’.Z’
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8. De Morgan’s Theorems Exercises
• Apply De Morgan’s theorems to the
expressions
–
–
–
–
–
–
–
–
(X’ + Y’ +Z’)’ = X’’.Y’’.Z’’ = X.Y.Z
((A + B + C)D)’ = (A + B + C)’ + D’
= A’.B’.C’ + D’
(ABC + DEF)’ = (ABC)’.(DEF)’
= A’ + B’ C’ . D’ + E’ + F’
(AB’ + C’D + EF)’ = (AB’)’.(C’D)’.(EF)’
= A’ + B’’.C’’ + D’.E’ + F’
= A’ + B.C + D’.E’ + F’
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9. De Morgan’s Theorems Exercises
• The Boolean expression for an ex-OR gate is
AB’ + A’B. Develop an expression for the exNOR gate
– Ex-OR = AB’ + A’B;
– Ex-NOR = (AB’ + A’B)’
= (AB’)’.(A’B)’
= (A’ + B’’) . (A’’ + B’)
= (A’ + B) . (A + B’)
= A’A + A’B’ + BA + BB’
= 0 + A’B’ + AB + 0
= A’B’ + AB
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10. Simplification using Boolean
Algebra
• The aim is to reduce the number of gates
used to implement a circuit
• Examples
– Simplify the following expressions using Laws
and Rules of Boolean Algebra, and De
Morgan’s Theorems where necessary
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11. Simplification using Boolean
Algebra - Examples
1. AB + A(B + C) + B(B + C)
AB + AB + AC + BB + BC
AB + AB + AC + B + BC
AB + AC + B + BC
AB + AC + B
B + AC
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