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Select the Boolean function(s) equivalent to x + yz, where x, y, and z are Boolean variables, and + denotes logical OR operation.
- a)x + z + xy
- b)(x + y)(x + z)
- c)x + xy + yz
- d)x + xz + xy
Correct answer is option 'B,C'. Can you explain this answer?
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Select the Boolean function(s) equivalent to x + yz, where x, y, and z...
Equivalent Boolean functions for x yz
To find the equivalent Boolean functions for the expression x yz, where x, y, and z are Boolean variables, we need to apply the rules of Boolean algebra. The expression x yz denotes the logical OR operation between x and yz.
Boolean Algebra Rules:
1. x + 0 = x (Identity Law)
2. x + 1 = 1 (Domination Law)
3. x + x' = 1 (Complement Law)
4. x + x = x (Idempotent Law)
5. x + y = y + x (Commutative Law)
6. x + (y + z) = (x + y) + z (Associative Law)
7. x + xy = x (Absorption Law)
8. x + x'y = x + y (Consensus Law)
Applying Boolean Algebra:
Given expression: x yz
We can break down the expression into two parts: x and yz. We need to find the equivalent Boolean functions for each part separately.
1. x:
- Since x is already a Boolean variable, the equivalent Boolean function for x is x itself.
2. yz:
- We can simplify yz using the distributive law: yz = (y + z)(y + z').
- Applying the identity law and the complement law, we can further simplify the expression:
yz = (y + z)(y + z') = yy + zy + yz + zz' = y + zy + yz + 0 = y + zy + yz.
Equivalent Boolean functions:
Using the above simplifications, we can express the given expression x yz as the sum of the equivalent Boolean functions for x and yz:
x yz = x + y + zy + yz.
Options:
The options given are:
a) x z xy
b) (x y)(x z)
c) x xy yz
d) x xz xy
Correct options:
From the above simplification, we can see that the equivalent Boolean functions for x yz are x + y + zy + yz.
Option B: (x y)(x z)
- This option is equivalent to xx + yx + xz + yz, which is not the same as x + y + zy + yz.
- Therefore, option B is not correct.
Option C: x xy yz
- This option is equivalent to x + xy + yz, which is the same as x + y + zy + yz.
- Therefore, option C is correct.
Therefore, the correct options are B and C.
To find the equivalent Boolean functions for the expression x yz, where x, y, and z are Boolean variables, we need to apply the rules of Boolean algebra. The expression x yz denotes the logical OR operation between x and yz.
Boolean Algebra Rules:
1. x + 0 = x (Identity Law)
2. x + 1 = 1 (Domination Law)
3. x + x' = 1 (Complement Law)
4. x + x = x (Idempotent Law)
5. x + y = y + x (Commutative Law)
6. x + (y + z) = (x + y) + z (Associative Law)
7. x + xy = x (Absorption Law)
8. x + x'y = x + y (Consensus Law)
Applying Boolean Algebra:
Given expression: x yz
We can break down the expression into two parts: x and yz. We need to find the equivalent Boolean functions for each part separately.
1. x:
- Since x is already a Boolean variable, the equivalent Boolean function for x is x itself.
2. yz:
- We can simplify yz using the distributive law: yz = (y + z)(y + z').
- Applying the identity law and the complement law, we can further simplify the expression:
yz = (y + z)(y + z') = yy + zy + yz + zz' = y + zy + yz + 0 = y + zy + yz.
Equivalent Boolean functions:
Using the above simplifications, we can express the given expression x yz as the sum of the equivalent Boolean functions for x and yz:
x yz = x + y + zy + yz.
Options:
The options given are:
a) x z xy
b) (x y)(x z)
c) x xy yz
d) x xz xy
Correct options:
From the above simplification, we can see that the equivalent Boolean functions for x yz are x + y + zy + yz.
Option B: (x y)(x z)
- This option is equivalent to xx + yx + xz + yz, which is not the same as x + y + zy + yz.
- Therefore, option B is not correct.
Option C: x xy yz
- This option is equivalent to x + xy + yz, which is the same as x + y + zy + yz.
- Therefore, option C is correct.
Therefore, the correct options are B and C.
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Community Answer
Select the Boolean function(s) equivalent to x + yz, where x, y, and z...
Concept:
Boolean function:
The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labeled individually to represent inputs to the expression.
For example, variables A, B, C, etc, give us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.
The given expression is = x + yz the equivalent expression is,
Option 1: x + z + xy
= x + z + xy
= x(1+y)+ z (∴ 1 + x = x + 1 = x)
= x + z it is not equivalent x + yz.
Option 2: (x + y)(x + z)
= (x + y)(x + z)
= x + xz + yx + yz
= x(1+ z+ y) + yz (∴1 + x = x + 1 = x)
= x + yz it is equivalent x + yz.
Option 3: x + xy + yz
= x + xy + yz
= x(1 + y) +yz
= x + yz it is equivalent x + yz.
Option 4: x + xz + xy
= x + xz + xy
= x(1+z+y) (∴1 + x = x + 1 = x)
= x it is not equivalent x + yz.
Hence the correct answer is option 2 and option 3.
Boolean function:
The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labeled individually to represent inputs to the expression.
For example, variables A, B, C, etc, give us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1.
The given expression is = x + yz the equivalent expression is,
Option 1: x + z + xy
= x + z + xy
= x(1+y)+ z (∴ 1 + x = x + 1 = x)
= x + z it is not equivalent x + yz.
Option 2: (x + y)(x + z)
= (x + y)(x + z)
= x + xz + yx + yz
= x(1+ z+ y) + yz (∴1 + x = x + 1 = x)
= x + yz it is equivalent x + yz.
Option 3: x + xy + yz
= x + xy + yz
= x(1 + y) +yz
= x + yz it is equivalent x + yz.
Option 4: x + xz + xy
= x + xz + xy
= x(1+z+y) (∴1 + x = x + 1 = x)
= x it is not equivalent x + yz.
Hence the correct answer is option 2 and option 3.
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Select the Boolean function(s) equivalent to x + yz, where x, y, and z are Boolean variables, and + denotes logical OR operation.a)x + z + xyb)(x + y)(x + z)c)x + xy + yzd)x + xz + xyCorrect answer is option 'B,C'. Can you explain this answer?
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