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 Title Name Discrete Mathematics Solved Assignment for Amity better grades and marks in Delhi Noida Mumbai Faridabad and Ghaziabad India University AMITY Service Type Assignment Course B.C.A Semester Semester-II Course: B.C.A Short Name or Subject Code Discrete Mathematics Commerce line item Type Semester-II Course: B.C.A Product Assignment of B.C.A Semester-II (AMITY) Price PRICE INR Click to View Price

## Questions:-

1. (a) Explain Tautologies and Contradiction with help of example.

Definition: A propositional expression is a tautology if and only if for all possible assignments of truth values to its variables its truth value is T

Example: P V ¬ P

P     ¬ P     P V ¬ P

---------------------

T      F        T

F      T        T

Definition: A propositional expression is a contradiction if and only if for all possible assignments of truth values to its variables its truth value is F

Example: P Λ ¬ P

P     ¬ P     P Λ ¬ P

---------------------

T      F        F

F      T        F

Usage of tautologies and contradictions - in proving the validity of arguments; for rewriting expressions using only the basic connectives.

(b) Prove that the following propositions are Tautology

• p V ~p (ii) ~(p ^ q) V q    (iii) p => (p V q)

• P V ¬ P

A propositional expression is a tautology if and only if for all possible assignments of truth values to its variables its truth value is T

 P (P ∨ ¬P) F T T T
• ~(p ^ q) V q

 p q (¬(p ∧ q) ∨ q) F F T F T T T F T T T T
• p => (p V q)
 p q (p → (p ∨ q)) F F T F T T T F T T T T
1. Prove that with help of Boolean algebra
• (a+b)’ =a’.b’
• (ii) (a.b)’ =a’ + b’
1. Explain all logic gates with symbol and truth table. Simplyfy the Boolean expression

A+B(A+B)+A(A’+B)

Logic gates

Digital systems are said to be constructed by using logic gates. These gates are the AND, OR, NOT, NAND, NOR, EXOR and EXNOR gates. The basic operations are described below with the aid of truth tables.

AND gate

The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high.  A dot (.) is used to show the AND operation i.e. A.B.  Bear in mind that this dot is sometimes omitted i.e. AB

OR gate

The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high.  A plus (+) is used to show the OR operation.

NOT gate

The NOT gate is an electronic circuit that produces an inverted version of the input at its output.  It is also known as an inverter.  If the input variable is A, the inverted output is known as NOT A.  This is also shown as A´, or A with a bar over the top, as shown at the outputs. The diagrams below show two ways that the NAND logic gate can be configured to produce a NOT gate. It can also be done using NOR logic gates in the same way.

NAND gate

This is a NOT-AND gate which is equal to an AND gate followed by a NOT gate.  The outputs of all NAND gates are high if any of the inputs are low. The symbol is an AND gate with a small circle on the output. The small circle represents inversion.

NOR gate

This is a NOT-OR gate which is equal to an OR gate followed by a NOT gate.  The outputs of all NOR gates are low if any of the inputs are high.

The symbol is an OR gate with a small circle on the output. The small circle represents inversion.

EXOR gate

The ´Exclusive-OR´ gate is a circuit which will give a high output if either, but not both, of its two inputs are high.  An encircled plus sign () is used to show the EOR operation.

EXNOR gate

The ´Exclusive-NOR´ gate circuit does the opposite to the EOR gate. It will give a low output if either, but not both, of its two inputs are high. The symbol is an EXOR gate with a small circle on the output. The small circle represents inversion.

A+B(A+B)+A(A’+B)

A+B(A+B)+A*B

A+BA+B+A*B

A+BA+B+A*B

A+BA+B*B

A+B*A+B

A+B

Truth table:

The Boolean expression is reduced to (A + B)

Case Detail

1.

a => b= 1

a=> c=1

b=> d= 1

d=> c= 1

a     b    c     d

a  0     1    1     0

b  1     0    0     1

c  1     0    0     1

d  0     1    1     0

1. Simplify the following Boolean function F(A,B,C,D)= Σ( 0,1,2,3,4,5,7,6,8,9,11)

Karnaugh Map

 AB 00 01 11 10 CD 00 1 1 1 1 01 1 1 1 1 11 0 0 0 0 10 1 1 1 0

Truth Table

 A B C D F(ABCD) 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

So required equation is

 F(ABCD) = C ‘+ A’ D ‘+ B D ‘

Assignment C

1. Complement the expression by applying De-Morgan’s theorem (A+A’B)’?

Options

1. AB
2. A’B´
3. A’B’ **
4. AB’
1. Find the minimum sum of product using K-Map?

F= AB+AB’C+ABC

Options

1. F=A’B+AC
2. F=AB+AC’
3. F=AB’+AC
4. F=AB+AC**

3- what is the value of Boolean expression a+a’=?

Options

1. 1**
2. 2
3. a
4. a’
1. Consider the following--

p: Anil is rich                    q: Kanchan is poor

What is the symbolic form of the following statement?

Anil is poor and Kanchan is rich

Options

1. ~p ^ q
2. p ^ q**
3. ~p ^ ~q
4. None of above
1. Consider the following

P: This computer is good                    q: This computer is cheap

What is the symbolic form of the following statement?

This computer is neither good nor cheap

Options

1. ~p ^ q
2. p ^ ~q
3. (~p) ^ (~q)**
4. None of above
1. In which of the following gates the output is 1, if and only if at least one input is 1 ?

Options

1. NOR
2. AND
3. OR**
4. NAND
1. In which of the following gates the output is 0, if and only if at least one input is 1 ?

Options

1. NOT
2. AND
3. NOR**
4. NAND
1. What is the minimum number of two input NAND gates used to perform the function of two inputs OR gate?

Options

1. Three**
2. Two
3. One
4. Four
1. Which of the following gates are added to the inputs of the OR gate to convert it to the NAND gate?

Options

1. NOT**
2. AND
3. AND
4. XOR
1. What logic function is produced by adding an inverter to the output of an AND gate?

Options

1. NAND**
2. NOR
3. XOR
4. OR
1. Which of the following Boolean algebra expressions is incorrect?

Options

1. A+ A’+= A+B
2. A+AB =B **
3. (A+B)(A+B) = A+BC
4. (A+B’)(A+B)=A
1. The simplified form of the Boolean expression (X+Y+XY)(X+Z) is ?

Options

1. X+Y+Z
2. XY+YZ
3. X+YZ**
4. XZ+Y
1. The simplified form of the Boolean expression (X+Y’+Z)(Z+Y’+Z’) (X+Y+Z) is ?

Options

1. X’Y+Z’
2. X+Y’Z**
3. X
4. XY+Z’
1. A full binary tree with n leaves contains--

Options

1. n nodes
2. log2n nodes
3. (2n-1) nodes **
4. 2n nodes
1. A full binary tree with non- leaf nodes contains--

Options

1. (2n+1) nodes **
2. log2n nodes
3. (n+1) nodes
4. 2n nodes
1. A complete graph with five vertices is--

Options

1. Non planar
2. planar
3. a non regular graph
4. a tree **
1. Which of the following statement is true?

Options

1. (A+ B)(A+C)=AC+BC
2. (A+ B)(A+C)=AB+C
3. (A+ B)(A+C)=A+BC **
4. (A+ B)(A+C)=AC+B
1. A non empty connected graph G is Eulerian if and only if its vertices are all of--

Options

1. Odd degre
2. Even degree**
3. Both (a) and (b)
4. None
1. A ------ is a closed path of none zero length that does not contain repeated edge.

Options

1. Path
2. Simple path
3. Circuit**
4. None
1. A ------ is a path that does not contain a repeated vertex.

Options

1. Path
2. Simple path**
3. Circuit
4. None
1. The maximum number of edges in any simple graph with n vertices is--

Options

1. n
2. n+1
3. n(n-1)/2**
4. None
1. A simple graph is said to be ------------ if every vertex in graph is connected with every other vertex

Options

1. null
2. regular
3. complete **
4. None
1. A graph in which all vertices are equal degree is called a --------- graph--

Options

1. null
2. regular**
3. complete
4. None
1. What is the value of Boolean expression a + (a.b) =?

Options

1. b
2. a**
3. ab
4. a’
1. What is the value of Boolean expression a + 1 =?

Options

1. 0
2. a
3. 1**
4. none
1. Consider the following--

P: Today is Tuesday                    q: It is raining       r: it is cold

Write in simple sentence for      -q => (r ^ p)

This computer is neither good nor cheap

Options

1. If today is Tuesday, then it is raining
2. If today is not Tuesday, then it is raining or it is cold
3. If it is not raining, then it is cold and today is Tuesday
4. None of these
1. If there exists at least one path between every pair of vertices in a graph, the graph is known as--

Options

1. Complete graph
2. Disconnected graph
3. Connected graph**
4. Euler graph
1. The length of Hamiltonian path (if exists) in a connected graph of n vertices is--

Options

1. n-1**
2. n
3. n+1
4. n/1
1. If each node in a tree has value greater than every value in its left subtree and has value less than every value in its right subtree , the tree is known as--

Options

1. Complete tree
2. Full binary tree
3. Binary search tree**
1. Which of the following statement is false ?

Options

1. A tree contains a cycle**
2. A tree with n nodes contains n-1 edges
3. A tree is a connected graph
4. None
1. Graph can be implemented using

Options

1. arrays
3. queue
4. all of these

32. Traversing a binary tree first root and then left and right subtrees called------traversal

Options

1. postorder
2. preorder**
3. inorder
4. all of these

33. A circuit is a connected graph which includes every vertex of the graph is known as

Options

1. Euler
2. Unicursal
3. Hamiltorium**
4. Clique
1. The number of vertices of odd degree in a graph is

Options

1. Always even**
2. Always odd
3. Either even or odd
4. Always zero
1. If a binary tree traversal in inorder, then numbers of the node are printed in----- order

Options

1. Ascending
2. Descending**
3. randomly
4. none of these
1. Which gate is known as universal gate

Options

1. NOT gate
2. AND gate
3. NAND gate**
4. XOR gate
1. is the value of Boolean expression (a .b)’ =?a

Options

1. a+b
2. a’+b’**
3. b
4. None of above
1. Find the complement of Boolean expression xy’ +x’y

Options

1. x+y
2. (x’+y).(x+z’)**
3. x+y’
4. None
1. A Karnaugh map in four variables is a square divided into ---

Options

1. 8
2. 12
3. 14
4. 16**
1. A tree with n nodes has ----- edges

Options

1. n/2
2. n-1**
3. n
4. n+1

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January 29, 2015