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Explain Tautologies and Contradiction with Help of Example. |
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| Short Name or Subject Code | Discrete Mathematics |
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Discrete Mathematics
(a) Explain Tautologies and Contradiction with help of an example.
Answer:
(b) Prove that the following propositions are Tautology
p V ~p (ii) ~(p ^ q) V q (iii) p => (p V q)
Answer:
3. Prove that with help of Boolean algebra
(a+b)’ =a’.b’
(ii) (a.b)’ =a’ + b’
4. Explain all logic gates with symbol and truth table. Simplyfy the Boolean expression
A+B(A+B)+A(A’+B)
Answer:
Case Detail
1.
2. Simplify the following Boolean function F(A,B,C,D)= Σ( 0,1,2,3,4,5,7,6,8,9,11)
Assignment C
1. Complement the expression by applying De-Morgan’s theorem (A+A’B)’?
Options
AB
A’B'
A’B’
AB’
2. Find the minimum sum of product using K-Map?
F= AB+AB’C+ABC
Options
F=A’B+AC
F=AB+AC’
F=AB’+AC
F=AB+AC
3. what is the value of Boolean expression a+a’=?
Options
1
2
a
a’
4. 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
~p ^ q
p ^ q
~p ^ ~q
None of above
5. 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
~p ^ q
p ^ ~q
(~p) ^ (~q)
None of above
6. In which of the following gates the output is 1, if and only if at least one input is 1 ?
Options
NOR
AND
OR
NAND
7. In which of the following gates the output is 0, if and only if at least one input is 1 ?
Options
NOT
AND
NOR
NAND
8. What is the minimum number of two input NAND gates used to perform the function of two inputs OR gate?
Options
Three
Two
One
Four
9. Which of the following gates are added to the inputs of the OR gate to convert it to the NAND gate?
Options
NOT
AND
AND
XOR
10. What logic function is produced by adding an inverter to the output of an AND gate?
Options
NAND
NOR
XOR
OR
11. Which of the following Boolean algebra expressions is incorrect?
Options
A+ A’+= A+B
A+AB =B
(A+B)(A+B) = A+BC
(A+B’)(A+B)=A
12. The simplified form of the Boolean expression (X+Y+XY)(X+Z) is ?
Options
X+Y+Z
XY+YZ
X+YZ
XZ+Y
13. The simplified form of the Boolean expression (X+Y’+Z)(Z+Y’+Z’) (X+Y+Z) is ?
Options
X’Y+Z’
X+Y’Z
X
XY+Z’
14. A full binary tree with n leaves contains--
Options
n nodes
log2n nodes
(2n-1) nodes
2n nodes
15. A full binary tree with non- leaf nodes contains--
Options
(2n+1) nodes
log2n nodes
(n+1) nodes
2n nodes
16. A complete graph with five vertices is--
Options
Non planar
planar
a non regular graph
a tree
17. Which of the following statement is true?
Options
(A+ B)(A+C)=AC+BC
(A+ B)(A+C)=AB+C
(A+ B)(A+C)=A+BC
(A+ B)(A+C)=AC+B
18. A non empty connected graph G is Eulerian if and only if its vertices are all of--
Options
Odd degree
Even degree
Both (a) and (b)
None
19. A ------ is a closed path of none zero length that does not contain repeated edge.
Options
Path
Simple path
Circuit
None
20. A ------ is a path that does not contain a repeated vertex.
Options
Path
Simple path
Circuit
None
21. The maximum number of edges in any simple graph with n vertices is--
Options
n
n+1
n(n-1)/2
None
22. A simple graph is said to be ------------ if every vertex in graph is connected with every other vertex
Options
null
regular
complete
None
23. A graph in which all vertices are equal degree is called a --------- graph--
Options
null
regular
complete
None
24. What is the value of Boolean expression a + (a.b) =?
Options
b
a
ab
a’
25. What is the value of Boolean expression a + 1 =?
Options
0
a
1
none
26. 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
If today is Tuesday, then it is raining
If today is not Tuesday, then it is raining or it is cold
If it is not raining, then it is cold and today is Tuesday
None of these
27. If there exists at least one path between every pair of vertices in a graph, the graph is known as--
Options
Complete graph
Disconnected graph
Connected graph
Euler graph
28. The length of Hamiltonian path (if exists) in a connected graph of n vertices is--
Options
n-1
n
n+1
n/1
29. 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
Complete tree
Full binary tree
Binary search tree
Threaded tree
30. Which of the following statement is false ?
Options
A tree contains a cycle
A tree with n nodes contains n-1 edges
A tree is a connected graph
None
31. Graph can be implemented using
Options
arrays
linked list
queue
all of these
32. Traversing a binary tree first root and then left and right subtrees called------traversal
Options
postorder
preorder
inorder
all of these
33. A circuit is a connected graph which includes every vertex of the graph is known as
Options
Euler
Unicursal
Hamiltorium
Clique
34. The number of vertices of odd degree in a graph is
Options
Always even
Always odd
Either even or odd
Always zero
35. If a binary tree traversal in inorder, then numbers of the node are printed in----- order
Options
Ascending
Descending
randomly
none of these
36. Which gate is known as universal gate
Options
NOT gate
AND gate
NAND gate
XOR gate
37. is the value of Boolean expression (a .b)’ =?a
Options
a+b
a’+b’
a.b
None of above
38. Find the complement of Boolean expression xy’ +x’y
Options
x+y
(x’+y).(x+z’)
x+y’
None
39. A Karnaugh map in four variables is a square divided into ---
Options
8
12
14
16
40. A tree with n nodes has ----- edges
Options
n/2
n-1
n
n+1