Chapter 1, con't., Combinatorial circuits

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Chapter 1, con't., Combinatorial circuits
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Basic logic gates

• An and gate
• An or gate
• A not gate

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Discrete StructuresCMSC 250Lecture 6
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Combining determining I/O relationship

• From a circuit to a Boolean formula
• p ? (q r)

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Draw a circuit for

• p, q r are inputs.
• Simplify before building the circuit.

p q r output
1 1 1 1
1 1 0 1
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 0
0 0 1 0
0 0 0 0
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Number conversions

• Different number system bases are used when
  convenient
• some commonly-used bases are 10 (decimal), 2
  (binary), 8 (octal), 16 (hexadecimal)
• the base tells how many different numerals are
  used
• the base also determines the value of each place
• Conversions from anything to base 10
• use the definition of the number system
• Conversions from base 10 to anything
• use repeated integer division

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Addition of binary numbers

• Carry if the number would be too large for the
  number system- if it is greater than 1

1001
10
1001
11
1011
10
1101
111
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Discrete StructuresCMSC 250Lecture 7
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Addition of octal and hexadecimal numbers

• Carry if the number would be too large for the
  number system (larger than 7 or 15)

7238
128
2658
338
ABC16
1216
CDE16
ED16
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Two's complement

• To represent negative values in binary
• Find the binary equivalent of the absolute value.
• Pad on the left to completely fill the bits in
  the specified bit width
• Switch all of the 1's to 0's and 0's to 1's.
• Add 1 to the result.
• Example find the 8-bit two's complement
  representation of -43
• 4310 1010112
• 001010112
• 110101002
• 110101012 -4310

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Using a circuit for adding two bits
input
output
p q carry sum
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0

• Write as a logic expression
• Translate to circuits

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Half adder

• p and q are binary values (1 bit each)
• sum (p ? q) (p q)
• carry (p q)

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Full adder (for three bits)

• p, q, and r are binary digits
• p q r produces a sum value and a carry value

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Parallel adders

• Chain these half adders and full adders together
  for multi-bit addition
• X1X2X3 Y1Y2Y3 CA1A2A3

X3
Y3
X2
Y2
X1
Y1
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Discrete StructuresCMSC 250Lecture 8
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Draw a circuit for
a b c output
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 1
0 1 0 0 0 0 1 0 0
0 1 1 0 0 1 0 0 1
1 0 0 0 1 0 0 0 0
1 0 1 0 1 1 0 0 1
1 1 0 1 0 0 1 0 0
1 1 1 1 1 0 0 0 1
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Topic not covered

• Simplifying circuits- there are techniques which
  exist (which are complex).