Convert the Boolean expression to SOP form: AB +B(CD+EF)

1-Convert the Boolean expression to SOP form: AB +B(CD+EF)

2-Convert the Boolean expression to standard SOP form: AB +ĀBD+ĀCD 3-Express F = % ỹ + x Z in a standard (POs) form, then write F in conical (E T) forms. Draw the logic circuit of F.

The correct answer and explanation is :

Let’s break each part down carefully and provide a thorough explanation.

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1. Convert the Boolean expression to Sum of Products (SOP) Form:

Given expression:
[
AB + B(CD + EF)
]

Expanding using distribution:
[
AB + BCD + BEF
]
This is already in SOP form because it is a sum of product terms.

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2. Convert the Boolean expression to Standard SOP Form:

Given expression:
[
AB + \bar{A}BD + \bar{A}CD
]

Each term should have all variables (A, B, C, and D) explicitly present.

• AB lacks C and D → Expand using C + C’ and D + D’:
  [
  AB = ABCD + ABC\bar{D} + AB\bar{C}D + AB\bar{C}\bar{D}
  ]
• (\bar{A}BD) lacks C → Expand using C + C’:
  [
  \bar{A}BD = \bar{A}BCD + \bar{A}B\bar{C}D
  ]
• (\bar{A}CD) lacks B → Expand using B + B’:
  [
  \bar{A}CD = \bar{A}BCD + \bar{A}\bar{B}CD
  ]

Now, collecting all minterms:
[
ABC\bar{D} + AB\bar{C}D + AB\bar{C}\bar{D} + \bar{A}BCD + \bar{A}B\bar{C}D + \bar{A}\bar{B}CD
]
This is the standard SOP form.

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3. Express ( F = x \bar{y} + xz ) in Standard POS and Canonical Form

Given function:
[
F = x\bar{y} + xz
]

Convert to POS (Product of Sums) form

To convert to POS, use De Morgan’s theorem and distributive laws:

1. Complement the function using De Morgan’s Theorem:
   [
   F’ = (x\bar{y} + xz)’ = (x\bar{y})'(xz)’
   ]
2. Apply De Morgan’s theorem:
   [
   (x + y)(\bar{x} + \bar{z})
   ]
3. Complement again to get F:
   [
   F = (x + y)(\bar{x} + \bar{z})
   ]

Thus, the POS form is:
[
(x + y)(\bar{x} + \bar{z})
]

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Convert to Canonical (Σm) Form

Identify the minterms by writing a truth table.

x	y	z	F
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	1

Minterms where F = 1: ( m_4, m_5, m_6, m_7 )

[
F = Σm(4,5,6,7)
]

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Logic Circuit for F

The logic circuit consists of:

1. AND Gates for ( x\bar{y} ) and ( xz ).
2. OR Gate to combine them.

The circuit:

• NOT gate for ( \bar{y} ).
• AND gate for ( x \cdot \bar{y} ).
• AND gate for ( x \cdot z ).
• OR gate combining the two outputs.