MTH202 Final Term Exam Subjective Type Questions Solved

Question: Explain Valid Arguments.
Answer: When some statement is said on the basis of a set of other statements, meaning that this statement is derived from that set of statements, this is called an argument. The formal definition is “an argument is a list of statements called “premises” (or assumptions or hypotheses) which is followed by a statement called the “conclusion.”
A valid argument is one in which the premises imply the conclusion.
1) It cannot have true premises and a false conclusion.
2) If its premises are true, its conclusion must be true.
3) If its conclusion is false, it must have at least one false premise.
4) All of the information in the conclusion is also in the premises.
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Question: What is the Difference between combinations and permutations?
Answer: When we talk of permutations and combinations in everyday talk we often use the two terms interchangeably. In mathematics, however, the two each have very specific meanings, and this distinction often causes problems
In brief, the permutation of a number of objects is the number of different ways they can be ordered; i.e. which one is first, which one is second or third etc. For example, you see, if we have two digits 1 and 2, then 12 and 21 are different in meaning. So their order has its own importance in permutation.
On the other hand, in combination, the order is not necessary. you can put any object at first place or second etc. For example, Suppose you have to put some pictures on the wall, and suppose you only have two pictures: A and B.
You could hang them
or

We could summarise permutations and combinations (very simplistically) as
Permutations - position important (although choice may also be important)
Combinations - chosen important,
which may help you to remember
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Question: What is the use of kruskal's algorithn in our daily life?
Answer: The Kruskal’s algorithm is usually used to find minimum spanning tree i.e. the possible smallest tree that contains all the vertices. The standard application is to a problem like phone network design. Suppose, you have a business with several offices; you want to lease phone lines to connect them up with each other; and the phone company charges different amounts of money to connect different pairs of cities. You want a set of lines that connects all your offices with a minimum total cost. It should be a spanning tree, since if a network isn't a tree you can always remove some edges and save money. A less obvious application is that the minimum spanning tree can be used to approximately solve the traveling salesman problem. A convenient formal way of defining this problem is to find the shortest path that visits each point at least once.

Question: What is irrational number?
Answer: Irrational number An irrational number can not be expressed as a fraction. In decimal form, irrational numbers do not repeat in a pattern or terminate. They "go on forever" (infinity). Examples of irrational numbers are: pi= 3.141592654...

Question: Define membership table and truth table.
Answer: Membership table: A table displaying the membership of elements in sets. Set identities can also be proved using membership tables. An element is in a set, a 1 is used and an element is not in a set, a 0 is used. Truth table: A table displaying the truth values of propositions.





Question: What is sequence and series?
Answer: Sequence A sequence of numbers is a function defined on the set of positive integer. The numbers in the sequence are called terms. Another way, the sequence is a set of quantities u1, u2, u3... stated in a definite order and each term formed according to a fixed pattern. U r =f(r) In example: 1,3,5,7,... 2,4,6,8,... 1 2 ,− 2 2 ,3 2 ,− 4 2 ,... Infinite sequence:- This kind of sequence is unending sequence like all natural numbers: 1, 2, 3, ... Finite sequence:- This kind of sequence contains only a finite number of terms. One of good examples are the page numbers. Series:- The sum of a finite or infinite sequence of expressions. 1+3+5+7+...
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Question: What is Euclidean algorithm?
Answer: In number theory, the Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (GCD) of two integers.
Its major significance is that it does not require factoring the two integers, and it is also significant in that it is one of the oldest algorithms known, dating back to the ancient Greeks.

Question: what is the circle definition?
Answer: A circle is the locus of all points in a plane which are equidistant from a fixed point. The fixed point is called centre of that circle and the distance is called radius of that circle

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Question: What is meant by Discrete?
Answer:
A type of data is discrete if there are only a finite number of values possible. Discrete data usually occurs in a case where there are only a certain number of values, or when we are counting something (using whole numbers). For example, 5 students, 10 trees etc.


Question: What are digital circuits?
Answer: Digital circuits are electric circuits based on a number of discrete voltage levels.
In most cases there are two voltage levels: one near to zero volts and one at a higher level depending on the supply voltage in use. These two levels are often represented as L and H.



Question: What is absurdity or contradiction?
Answer: A statement which is always false is called an absurdity.

Question: What is contingency?
Answer: A statement which can be true or false depending upon the truth values of the variables is called a contingency.
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Question: Is there any particular rule to solve Inductive Step in the mathematical Induction?
Answer: In the Inductive Step, we suppose that the result is also true for other integral values k. If the result is true for n = k, then it must be true for other integer value k +1 otherwise the statement cannot be true.
In proving the result for n = k +1, the procedure changes, as it depends on the shape of the given statement.
Following steps are main:
1) You should simply replace n by k+1 in the left side of the statement.
2) Use the supposition of n = k in it.
3) Then you have to simplify it to get right side of the statement. This is the step,
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where students usually feel difficulty.
Here sometimes, you have to open the brackets, or add or subtract some terms
or take some term common etc. This step of simplification to get right side of the given statement for n = n + 1 changes from question to question.
Now check this step in the examples of the Lessons 23 and 24.



Question: Different notations of conditional implication.
Answer: If p than q. P implies q. If p , q. P only if q. P is sufficient for q.

Question: What is cartesion product?
Answer: Cartesian product of sets:- Let A and B be sets. The Cartesian product of A and B, denoted A x B (read “A cross B”) is the set of all ordered pairs (a, b), where a is in A and b is in B. For example: A = {1, 2, 3, 4, 5, 6} B = {a} A x B = {(1,a), (2,a), (3,a), (4,a), (5,a)}

Question: Define fraction and decimal expansion.
Answer: Fraction:- A number expressed in the form a/b where a is called the numerator and b is called the denominator. Decimal expansion:- The decimal expansion of a number is its representation in base 10 The number 3.22 3 is its integer part and 22 is its decimal part The number on the left of decimal point is integer part of the number and the number on the right of the decimal point is decimal part of the number.
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Question: Explane venn diagram.
Answer: Venn diagram is a pictorial representation of sets. Venn diagram can sometime be used to determine whether or not an argument is valid. Real life problems can easily be illustrate through Venn diagram if you first convert them into set form and then in Venn diagram form. Venn diagram enables students to organize similarities and differences visually or graphically. A Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common

Question: Write the types of functions.
Answer: Types of function:- Following are the types of function 1. One to one function 2. Onto function 3. Into function 4. Bijective function (one to one and onto function) One to one function:- A function f : A to B is said to be one to one if there is no repetition in the second element of any two ordered pairs. Onto function:- A function f : A to B is said to be onto if Range of f is equal to set B (co-domain). Into function:- A function f : A to B is said to be into function of Range of f is the subset of set B (co domain) Bijective function: Bijective function:- A function is said to be Bijective if it is both one to one and onto.

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Question: Explain combinatorics.
Answer: Branch of mathematics concerned with the selection, arrangement, and combination of objects chosen from a finite set.
The number of possible bridge hands is a simple example; more complex problems include scheduling classes in classrooms at a large university and designing a routing system for telephone signals. No standard algebraic procedures apply to all combinatorial problems; a separate logical analysis may be required for each problem.

Question: How the tree diagram use in our real computer life?
Answer: Tree diagrams are used in data structure, compiler construction, in making algorithms, operating system etc.

Question: Write detail of cards.
Answer: Diamond Club Heart Spade A A A A 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 J J J J Q Q Q Q K K K K Where 26 cards are black & 26 are red. Also ‘A’ stands for ‘ace’ ‘J’ stands for ‘jack’ ‘Q’ stands for ‘queen’ ‘K’ stands for ‘king’

Question: what is the purpose of permutations?
Answer: Definition:- Possible arrangements of a set of objects in which the order of the arrangement makes a difference. For example, determining all the different ways five books can be arranged in order on a shelf. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. Purpose of permutation is to establish significance without assumptions

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By ADEEL ABBAS, Bhakkar. AdeelAbbasbk@gmail.com