K-Map MCQ Quiz - Objective Question with Answer for K-Map - Download Free PDF

F (P, Q, R, S) = ∑ 0, 2, 5, 7, 8, 10, 13, 15

Don’t care min terms are 2, 7, 8, 13

By plotting the K-map, the minimal SOP (sum of products) can be found.

Explanation –

While putting the terms to k-map following things happen,

• 3^rd and 4^th columns are swapped
• 3^rd and 4^th rows.
• term 2 is going to (0, 3) column instead of (0, 2)
• 8 is going to (3, 0) instead of (2,0)

Solving, the above K-map, we get Q̅S̅ + QS

Concept:

We are given a Boolean function in the nested form: \( f(f(xy, y), z) \). To simplify this, we need to understand how the function \( f \) behaves and use Boolean algebra rules.

Step 1: Assume definition of function

Let us assume \( f(a, b) = a + b \), a common definition used in Boolean simplification unless otherwise specified.

So,

\( f(xy, y) = xy + y \)

Using the identity: \( xy + y = y \) (Since \( y(x + 1) = y \))

Step 2: Substitute into outer function

Now substitute into the outer function:

\( f(y, z) = y + z \)

Hence, the simplified expression is:

\( y + z \)

Compare with Options:

• Option 1: \( \bar{x} + z \) ❌
• Option 2: \( xyz \) ❌
• Option 3: \( xy̅ + z \) ❌
• Option 4: None of the options ✅

Final Answer: ✅ None of the options

The correct answer is : A̅ B̅ + C

Explanation:

To find the minimum product of sums (POS) for the given Karnaugh map (K-map), we'll follow these steps:

1. Analyze the K-map:

• Identify groups of 1s and determine their corresponding sum terms.
• Identify groups of 0s and determine their corresponding product terms.
• We'll use the don't-care conditions (marked as 'x') to simplify the expression.

Given: \(\rm Y=f(ABCD)=\Sigma(0,1,2,6,7,10,14)+\Sigma d(3,8,11,15)\)

Grouping the given equation we will get the boolean function as: Y = A̅ B̅ + C

The correct answer is (B), (D), (A), (C).

Key Points

• Draw the K-map for the given Boolean function (B): The first step in simplifying a Boolean function using a Karnaugh map (K-map) is to draw the K-map. This involves setting up a grid based on the number of variables in the function.
• Transfer the truth table values to the K-map (D): Next, the truth table values are transferred to the K-map. Each cell in the K-map corresponds to a specific combination of input variables, and the values from the truth table are placed in the appropriate cells.
• Identify and group the largest possible cluster of 1's (A): After transferring the values, the next step is to identify and group the largest possible clusters of 1's in the K-map. These clusters help in simplifying the Boolean expression.
• Write the simplified Boolean expression from the grouped clusters (C): Finally, the simplified Boolean expression is written based on the grouped clusters of 1's. This step involves deriving the simplified terms from the clusters.

Thus the correct answer is ​(B), (D), (A), (C).

Additional Information

• Karnaugh map (K-map): A Karnaugh map is a diagram used in Boolean algebra and digital logic design for the simplification of algebraic expressions. The K-map provides a visual method of grouping expressions with common factors and eliminating unwanted variables.
• Simplification Process: The simplification process using a K-map helps in minimizing the number of logical operations required, which in turn reduces the complexity of digital circuits.
• Grouping Clusters: Grouping the largest possible clusters of 1's is crucial as it ensures that the simplest form of the Boolean expression is achieved. Larger clusters lead to fewer terms in the simplified expression.

The correct answer is (a + b̅ + c) (a + c̅ + d) (a̅ + b)

Explanation:

To find the minimum product of sums (POS) for the given Karnaugh map (K-map), we'll follow these steps:

1. Analyze the K-map:

• Identify groups of 1s and determine their corresponding sum terms.
• Identify groups of 0s and determine their corresponding product terms.
• We'll use the don't-care conditions (marked as 'x') to simplify the expression.

Given K-map:

Step-by-Step Simplification:

=> [(a + b + c̅ + d). (a + b̅ + c̅ + d)] [(a + b̅ + c + d) (a + b̅ + c +  d̅)] [(a̅ + b + c + d) (a̅ + b + c + d̅) (a̅ + b + c̅ + d̅) (a̅ + b + c̅ + d)]

=> [(a + c̅ + d) + (b̅.b)] [(a + b̅ + c) + (d̅.d)] [((a̅ + b + c) + (d̅.d)) ((a̅ + b + c̅) + (d̅.d))]   :  a̅.a = 0
=> [(a + c̅ + d) + 0] [(a + b̅ + c) + 0] [((a̅ + b + c) + 0) ((a̅ + b + c̅) + 0)]   

=> (a + c̅ + d) (a + b̅ + c) [(a̅ + b + c) (a̅ + b + c̅)]   

=> (a + c̅ + d) (a + b̅ + c) [(a̅ + b) + c̅.c]   :  a̅.a = 0

=> (a + c̅ + d) (a + b̅ + c) [(a̅ + b) + 0]

=> (a + c̅ + d) (a + b̅ + c) (a̅ + b)

Simplified POS:​
The minimal product of sums expression is: (a + b̅ + c) (a + c̅ + d) (a̅ + b)

Therefore, the correct option is: 4) (a + b̅ + c) (a + c̅ + d) (a̅ + b)

The correct answer is option 1.

Concept:

The given K-Map is,

F = a'b'c+a'bc'+ab'c'+abc

F= a'(b'c+bc')+a (b'c'+bc)

F=a'(b⊕c)+a(b⊙c)

F=a'(b⊕c)+a(b⊕c)'

F=a⊕b⊕c

Hence the correct answer is Exclusive OR.

The correct answer is option 4

K-maps

F(A, B, C, D) = ∑ (0, 1, 2, 3, 6, 12, 13, 14, 15)

Two K-Maps can be constructed from the given boolean function

The expression for K-Map 1 is AB + A'B' + A'CD'

The expression for K-Map 2 is AB +A'B' + BCD'

Concept:

Draw the K- map, convert the K-map into a SOP (sum of product) or POS (product of sum) form. While reducing the K-map in these forms, a don’t care will be needed only when with the use of don’t cares we can reduce the term size.

Diagram: K – Map

From the K-map simplification:

F(a, b, c, d) = a̅.c

F(a, b, c, d) = \(\overline {\left( {a + \bar c} \right)}\)

Diagram:

Therefore, only one NOR gate is needed to implement the minimized function

Concept:

We can simplify the given boolean function with the help of K-Map.

The K-map is a systematic way of simplifying Boolean expressions. With the help of the K-map method, we can find the simplest POS and SOP expression, which is known as the minimum expression.

Just like the truth table, a K-map contains all the possible values of input variables and their corresponding output values.

The K-map method is used for expressions containing 2, 3, 4, and 5 variables.

Calculation:

Given ;  F (x, y, z) = Σ(0, 2, 4, 5, 6) 

3 variable K-map:

The grouping of cells has shown below

The expression obtained from the K-Map → F = z' + xy'

Additional Information Note 1 − If outputs are not defined for some combination of inputs, then those output values will be represented with the don’t care symbol ‘x’. That means, we can consider them as either ‘0’ or ‘1’.

Note 2 − If don’t care terms are also present, then place doesn’t care ‘x’ in the respective cells of the K-map. Consider only the don’t care ‘x’ that are helpful for grouping the maximum number of adjacent ones. In those cases, treat the don’t care value as ‘1’.

F(P, Q, R, S) = PQ + P̅QR + P̅QR̅S

F(P, Q, R, S) = PQ(R + R̅)(S + S̅) + P̅QR(S + S̅) + P̅QR̅S  

F(P, Q, R, S) = PQRS + PQRS̅ + PQR̅S + PQR̅S̅ + P̅QRS + P̅QRS̅ + P̅QR̅S

F(P, Q, R, S) = ∑ (15 + 14 + 13 + 12 + 7 + 6 + 5)  

K-Map:

F(P, Q, R, S) = PQ + QR + QS

Concept:

The K-map is a graphical method that provides a systematic method for simplifying and manipulating the Boolean expressions or to convert a truth table to its corresponding logic circuit in a simple, orderly process.

In an 'n' variable K map, there are 2n cells

For 4 variables there will be 24 = 16 cells as shown:

Calculations:

F(a, b, c) = ∑(0, 2, 4, 5, 6)

F = c' + ab'

Karnaugh map (K-map):

• The Karnaugh map (K-map) is a method of simplifying Boolean algebra expressions.
• The Karnaugh map reduces the need for extensive calculations.
• Karnaugh map can be explained as An array that contains 2k number of cells, where k is the number of variables in the Boolean expression that is to be reduced or optimized.

For 4 variables there will be 24 = 16 cells as shown:

Number of cells in 2 variable k-map = 22 = 4

Number of cells in 3 variable k-map = 23 = 8

Number of cells in 4 variable k-map = 24 = 16

Number of cells in 5 variable k-map = 2⁵ = 32

The “Don’t Care” conditions allow us to replace the empty cell of a k-map and form a grouping of the variables which is larger than that of original groups. 

While forming groups of cells, we can consider a “Don’t Care” cell as 1 or 0 or we can also ignore that cell. 

Let us assume that A, B, C, and D are the 4 variables.

K- map for given function:

f(A, B, C, D) = BD + B̅ D̅

Implementing the above function by using XOR gates with complements of variables:

\(f\left( {A,\;B,\;C,\;D} \right) = \overline {\overline {B + \bar D} + \overline {\bar B + D} }\)

\(f\left( {A,\;B,\;C,\;D} \right) = \left( {B + \bar D} \right).\left( {\bar B + D} \right)\)

f(A, B, C, D) = BD + B̅ D̅

Explanation:-

As we already know from 'k' variables, (2^k) numbers can be formed. Thus the number of possible functions with 'k' variables as input such that there are 'm' minterms are,

2^kC_m or C(2^k, m)

Where (2^k ) is a possible number from 'k' variables and 'm' are the desired number of minterm for which the number of the function needs to be calculated. 

For example,

Counting the number of Boolean functions possible with two variables such that there are exactly two minterms.

Calculation:-

As we already know from two variables (a and b) four numbers (0, 1, 2, 3) can be formed i.e, in binary digits 00, 01, 10, 11 and possible Min terms are a'b', 

a’b, ab’, ab respectively which gives ‘1’ as the output for respective binary digits as input.

Thus the number of possible functions with two variables as input such that there are exactly two minterms are,

⁴C₂ = 4! / 2!(4 - 2)! = 24 / (2 × 2) = 24/4 = 6

Where ‘4’ is the possible number from two variables and ‘2’ is the desired number of minterms for which the number of the function needs to be calculated.

Explanation :

Given:

f(w, 0, 0, z)= 1

f(1, x, 1, z)= x+z

f(w, 1, y, z)= wz+y

Only thing makes this Question complicated is how you fill the K map cells.

1. f(w, 0, 0, z) = 1 ; means whenever x = 0 and y = 0 (total 4 cells) then put 1 in k-map cell.

2. f(1, x, 1, z) = x+z ; means whenever w=1 and y=1 (total 4 cells) then evaluate the expression “x+z” as per each cell (only those where w=1 and y=1) and fill the result in that particular cell.

3. f(w, 1, y, z) = wz+y ; means whenever x=1( total 8 cells) then evaluate the expression “wz+y” as per each cell (only those where x=1) and fill the result in that particular cell.

1. In the Below K map first we mark x = 0 and y = 0

  1001, 0000 , 0001 , 1000 these 4 cells will be marked as 1 .

  (wx’y’z , w’x’y’z’ , w’x’y’z , wx’y’z’) likewise we can fill the case where w=1 and y=1.

2. Now x=1( total 8 cells)

                     0100,     0101,           0110,          0111,          1100,        1101,        1110,        1111

                     w’xy’z’,      w’xy’z,      w’xyz’,        w’xyz,        wxy’z’,      wxy’z,        wxyz’,       wxyz

Likewise we can fill the K map as given in Image.

So after minimum sum of product expression will be x’y’ + xy + wz ; so total 6 literals. Answer will be 6.