MTH202 Final Term Solved Subjective Lectures 23-45 with References

MTH202 Discrete Mathematics

Lecture Wise Questions and Answers

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Q. What is inductive step?

Answer: In basis step we check that the given statement or proposition is true for n=1 and in Inductive step we first suppose that the proposition is true for n=k and prove that it is also true for n=k+1. In the inductive step we suppose that a function is true for an arbitrary integer k and then we prove that the given statement is also hold for k+1. It means that the statement is true for all positive integers n Q. How can we define propositional function ?

Answer: In simple words propositional function is an expression in form of proposition but having undefined symbols (variables) and become a proposition by assigning appropriate values to these symbols. Q. In the Basis step of mathematical Induction, I've seen in some problems P(0) is used while in some P(1) is used.

In the basic definition we've to find P(1) is true. So what's the reason of using P(0)?

Answer: Principle of Mathematical Induction is defined as follow: Let P(n) be a property that is defined for integers n, and let ' a ' be a fixed integer.

Suppose the following two statements are true:

1. P(a) is true.
2. For all integers k ≥ a, if P(k) is true then P(k + 1) is true.

Then the statement for all integers n ≥ a, P(n) is true.

Here P(1) is not fixed, we take P(a) for some fixed ' a ' and than suppose for some k ≥ a, and so on. Q. Explain the last exercise of the lecture ( DeMorgan's Law )

Answer: That is the DeMorgan's Law for the n sets. For understanding it, you should see first DeMorgan's Law for two sets A and B which is following: ( A U B )c = Ac ∩ Bc

For n number of sets A1, A2, ..., An, DeMorgan's Law is the following:

( A1 U A2 U ... U An )c = A1 c ∩ A2 c ∩ .... ∩ An c

This is to be proved using Mathematical Induction in the lecture. Q.In first example basic step why it is written that series is ending at one while interger are increasing?

Answer: In the Mathematical Induction, we take some initial value of n in the basic step. This initial value can be 0 or 1.

In the example, we take n = 1 in the series. It means we take only one term in the series as an initial value for our Mathematical Induction.

Q.what is consecutive positive integer and inequality?

Consecutive integers are integers that follow each other in order. For Example 1,2,3,... are called consecutive positive integers. You can see, we will get next integer by adding 1 in the current integer. Inequality is a statement in which equal sign is not involved or which is not equal. For example: 2 < 5 or 7 > 4.

Q. What is the rule of direct proof?

In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two even integers is always even. Q. What is rational number and integer? The natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers.

A rational number is a number that can be in the form p/q where p and q are integers and q is not equal to zero. Any whole number is rational. Its denominator is 1. For instance, 8 equals 8/1, which is the quotient of two integers. A number like square root of 16 is rational, since it can be expressed as the quotient of two integers in the form 4/1. The following are also examples of rational numbers: 5/6,-6/7,3/1 etc Q. Define Prime number? A prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Diviser is the number that can divide your desired number. e.g. 6 has the following divisor 1,2,3,6.

Q. What is Direct Method and Indirect method?

There are two methods of proof Direct Method and Indirect Methods. In Direct Method we try to prove the statement directly as it is . For example " Prove that the sum of two odd integers is even." We will take two odd integers and try to prove that their sum is even. While in Indirect Methods we try to restate the given statement and use this to prove our objective. Q. Define Geometric Sequence or Geometric Progression (G.P.)? A sequence in which every term after the first is obtained from the preceding term by multiplying it with a constant number is called a geometric sequence or geometric progression (G.P.) The constant number, being the ratio of any two consecutive terms is called the common ratio of the G.P. commonly denoted by “r”. Example: 6, -2, 2/3, - 2/9, … (common ratio = r = -1/3). The sum of the terms of a geometric sequence forms a geometric series (G.S.). In general, if a is the first term and r the common ratio of a geometric series, then the series is given as: a + ar + ar2 + ar3 + … For Example: 6 -2 + 2/3 - 2/9 +… (common ratio = r = -1/3) is geometric series. For finite series the formula is Sn= (a(1-rn) /(1-r)) since r ¹1. For the given example: 6 -2 + 2/3 - 2/9 +… (common ratio = r = -1/3) is an infinite geometric series so the formula to find sum is Sn= (a/(1-r)) since |r| < 1. Sn= (a/(r-1)) if |r| > 1.

Q. Define RECURSION?

It is the process of describing an action in terms of itself. OR The process of defining an object in terms of smaller versions of itself is called recursion. A recursive definition has two parts:

1. BASE:

An initial simple definition which cannot be expressed in terms of smaller versions of itself.

2. RECURSION:

The part of definition which can be expressed in terms of smaller versions of itself