Explain the three Boolean algebra laws using ORing and ANDing, and explain logic gates in terms of the number…

Question

Explain the three Boolean algebra laws using ORing and ANDing, and explain logic gates in terms of the number system.

ScanSolve AI Tutor · Accepted Answer

Here are the answers for questions 2, 3, and 4:

2.

a) Explain the three Boolean algebra laws you know using ORing and ANDing system.

Commutative Law: This law states that the order of operands does not affect the result of an operation.
- For ORing: $A + B = B + A$
- For ANDing: $A \cdot B = B \cdot A$
Associative Law: This law states that the grouping of operands does not affect the result of an operation when performing the same operation multiple times.
- For ORing: $(A + B) + C = A + (B + C)$
- For ANDing: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
Distributive Law: This law describes how operations interact with each other. It allows for the expansion or factorization of Boolean expressions.
- $A \cdot (B + C) = (A \cdot B) + (A \cdot C)$
- $A + (B \cdot C) = (A + B) \cdot (A + C)$

b) Explain logic gate in terms of number system.

A logic gate operates within the binary number system, which uses only two digits: 0 and 1. These digits represent the two possible states of an electrical signal (e.g., 0 for low voltage/off, 1 for high voltage/on). Logic gates take one or more binary inputs and produce a single binary output based on a specific Boolean function, effectively performing arithmetic or logical operations on these binary numbers.

3. Simplify the following Boolean algebra.

i) $A + \bar{A}B + A + B$

Apply the Idempotence Law ( $X+X=X$ ) for $A+A$ .
$A + \bar{A}B + A + B = A + \bar{A}B + B$
Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).
$A + \bar{A}B + B = A + B$
The simplified expression is $\boxed{A+B}$ .

ii) $Z = AB + A(B+C) + B(B+C)$
Apply the Distributive Law ( $X(Y+Z) = XY+XZ$ ).
$Z = AB + AB + AC + BB + BC$
Apply the Idempotence Law ( $X+X=X$ and $XX=X$ ).
$Z = AB + AC + B + BC$
Factor out B from terms containing B.
$Z = B(A+1) + AC + BC$
Apply the Identity Law ( $X+1=1$ ).
$Z = B(1) + AC + BC$ $Z = B + AC + BC$
Factor out B again from terms containing B.
$Z = B(1+C) + AC$
Apply the Identity Law ( $1+X=1$ ).
$Z = B(1) + AC$ $Z = B + AC$
The simplified expression is $\boxed{B+AC}$ .

iii) $F = A + \bar{A}BC + \bar{A}\bar{B}C$
Factor out $\bar{A}C$ from the last two terms.
$F = A + \bar{A}C(B + \bar{B})$
Apply the Complement Law ( $B + \bar{B} = 1$ ).
$F = A + \bar{A}C(1)$
Apply the Identity Law ( $X \cdot 1 = X$ ).
$F = A + \bar{A}C$
Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).
$F = A + C$
The simplified expression is $\boxed{A+C}$ .

4. Form the truth table using the Boolean expression of the following:

i) $F = A + \bar{A}BC + \bar{A}\bar{B}C$
From 3(iii), we simplified this expression to $F = A+C$ . We will use this simplified form to construct the truth table.

A C F = A+C
0 0 0
0 1 1
1 0 1
1 1 1

ii) $Z = AB + A(B+C) + B(B+C)$
From 3(ii), we simplified this expression to $Z = B+AC$ . We will use this simplified form to construct the truth table.

A B C AC Z = B+AC
0 0 0 0 0
0 0 1 0 0
0 1 0 0 1
0 1 1 0 1
1 0 0 0 0
1 0 1 1 1
1 1 0 0 1
1 1 1 1 1

Explain the three Boolean algebra laws using ORing and ANDing, and explain logic gates in terms of the number system.

Step-by-step solution

Apply the Idempotence Law ( $X+X=X$ ) for $A+A$ .

Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).

Apply the Distributive Law ( $X(Y+Z) = XY+XZ$ ).

Apply the Idempotence Law ( $X+X=X$ and $XX=X$ ).

Factor out B from terms containing B.

Apply the Identity Law ( $X+1=1$ ).

Factor out B again from terms containing B.

Apply the Identity Law ( $1+X=1$ ).

Factor out $\bar{A}C$ from the last two terms.

Apply the Complement Law ( $B + \bar{B} = 1$ ).

Apply the Identity Law ( $X \cdot 1 = X$ ).

Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).

Still stuck on this one?

More Computer Science Questions

A	C	F = A+C
0	0	0
0	1	1
1	0	1
1	1	1

A	B	C	AC	Z = B+AC
0	0	0	0	0
0	0	1	0	0
0	1	0	0	1
0	1	1	0	1
1	0	0	0	0
1	0	1	1	1
1	1	0	0	1
1	1	1	1	1

Explain the three Boolean algebra laws using ORing and ANDing, and explain logic gates in terms of the number system.

Step-by-step solution

Apply the Idempotence Law (X+X=XX+X=XX+X=X) for A+AA+AA+A.

Apply the Absorption Law (X+XˉY=X+YX + \bar{X}Y = X+YX+XˉY=X+Y).

Apply the Distributive Law (X(Y+Z)=XY+XZX(Y+Z) = XY+XZX(Y+Z)=XY+XZ).

Apply the Idempotence Law (X+X=XX+X=XX+X=X and XX=XXX=XXX=X).

Factor out B from terms containing B.

Apply the Identity Law (X+1=1X+1=1X+1=1).

Factor out B again from terms containing B.

Apply the Identity Law (1+X=11+X=11+X=1).

Factor out AˉC\bar{A}CAˉC from the last two terms.

Apply the Complement Law (B+Bˉ=1B + \bar{B} = 1B+Bˉ=1).

Apply the Identity Law (X⋅1=XX \cdot 1 = XX⋅1=X).

Apply the Absorption Law (X+XˉY=X+YX + \bar{X}Y = X+YX+XˉY=X+Y).

Still stuck on this one?

More Computer Science Questions

Apply the Idempotence Law ( $X+X=X$ ) for $A+A$ .

Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).

Apply the Distributive Law ( $X(Y+Z) = XY+XZ$ ).

Apply the Idempotence Law ( $X+X=X$ and $XX=X$ ).

Apply the Identity Law ( $X+1=1$ ).

Apply the Identity Law ( $1+X=1$ ).

Factor out $\bar{A}C$ from the last two terms.

Apply the Complement Law ( $B + \bar{B} = 1$ ).

Apply the Identity Law ( $X \cdot 1 = X$ ).

Apply the Absorption Law ( $X + \bar{X}Y = X+Y$ ).