MTH001 Midterm FAQs and Short Answers

Describe the Matrix representation of transitive relation?

We know that

1. If a relation is reflexive then its matrix representation has unit diagonal, to say as follow
2. If a relation is symmetric then its matrix representation is also symmetric about the diagonal, to say as follows;

Here you can see that matrix has same upper and lower triangular entries. Now question arise what kind of special structure or pattern we obtain in the case of transitive relation? Well!! It is not possible to find such kind of specialty in the pattern or structure of matrix representation of transitive relation. But in this case there should preserve (we get) a shortest “path”, means to say if we have (a,b) and (b,c) then definitely there exist (a,c). In other words if there are the “PATHS” from ab & ac then for the transitive relation there must exist ac. In matrix representation if we have 1,1 entries corresponding to (a,b) and (b,c) respectively then there must exist(we should “search” for) entry ‘1’ at (a,c) e.g: For a set say A= {a,b,c}, we have a relation say R1={(a,a),(a,b),(a,c),(b,c),(c,c)} A B C A 1 B 0 C 0

Here you can verify all the criteria of transitivity as discussed above.

What is difference between percent and percentage?

Percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or 0.45. Percentages are used to express how large one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 /

2. 50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.

Define decimal fraction and common fraction. A decimal fraction is a fraction where the denominator is a power of ten. e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as: 0.8, 0.83, 0.083, and 0.0008. A common fraction is a fraction in which numerator and denominator are both integers, as opposed to fractions. For example 2/5is a common fraction

Why we use round function?

The ROUND function is used to round a numeric field to the number of decimals specified. The Round function returns a number rounded to a specified number of decimal places. However, the Round function behaves a little peculiar, so before using this function, please read the following: The Round function utilizes round-to-even logic. If the expression that you are rounding ends with a 5, the Round function will round the expression so that the last digit is an even number. For example: Round (12.55, 1) would return 12.6 (rounds up) Round (12.65, 1) would return 12.6 (rounds down) Round (12.75,

1. would return 12.8 (rounds up)

What is the difference between a sequence and a series?

A sequence is a list of numbers. The order in which the numbers are listed is important, as 1, 2, 3, 4, 5... is one sequence, and 2, 1, 4, 3, 6, 5, ..is an entirely different sequence. A series is a sum of numbers. For example, 1 + 1/2 + 1/4 + 1/8 + 1/16 +... is an example of a series. A series is composed of a sequence of terms that are added up. The order in which the terms appear is sometimes important; there are certain kinds of series in which any rearrangement of the terms will not affect its sum (these are called "absolutely convergent"). There are others in which different arrangements of the terms will cause the sum of the series to change. Sequences and series are often confused because they are closely related to each other. But when we say "sequence," we are not concerned with the sum of the values of the terms, whereas in a series, we are interested in such a sum. For example: 1, 1/2, 1/3, 1/4, 1/5, 1/6,... is a sequence, and it is n(th) term will be 1/n

Define arithmetic operators. The arithmetic operators are • Addition • Subtraction • Multiplication • Division • Exponents

How does form relations (R1, R2, R3, R4...) on Cartesian product?

The Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y (e.g. the whole of the x-y plane): R=X × Y = {(x, y) / x e X and y e Y} then every possible subset of R is a relation denoted by R1, R2, R3…. For Example: X = {1, 2} and Y = {4, 5} R=X × Y = {(1, 4), (1, 5), (2, 4), (2, 5)} R1= {(1, 4)} R2= {(1, 5)} R3= {(2,

4. } R4= {(2, 5)} R5= {(1, 4), (1, 5)} R6= {(1, 4), (2, 4)} R7= {(1, 4), (2, 5)} R8= {(1, 5), (2, 4)} R9= {(1, 5), (2, 5)} R10= {(2, 4), (2, 5)} R11= {(1, 4), (1, 5), (2, 4)} R12= {(1,
5. , (1, 5), (2, 5)} R13= {(1, 4), (2, 4), (2, 5)} R14= {(1, 5), (2, 4), (2, 5)} R=R15 = {(1,
6. , (1, 5), (2, 4), (2, 5)} R16= { }.

Explain Venn diagram. 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

What is a function?

A function say 'f' is a rule or machine from a set A to the set B if for every element say a of A, there exist a unique element say b of set B such that b=f(a) Where b is the image of a under f, and a is the pre-image. Note it that set A is called the domain of f and Y is called the co domain of f. As we know that function is a rule or machine in which we put an input, and we get an output .Like that a juicer machine. We take some apples (here apples are input) and we apply a rule or a function of juicer machine on it, then we get the output in the form of juice

What is domain and co -domain. Domain means "the set of all x-coordinates in a relation". It is very simple, Let we take a function say f from the set X to set Y. Then domain means a set which contain all the elements of the set X. And co domain means a set which contain all the elements of the set Y. For example: Let we define a function "f" from the set X= {a, b, c, d} to Y= {1, 2, 3, 4}. such that f(a)=1, f(b)=2, f(c)=3, f(d)=1 Here the domain set is {a, b, c , d} And the co-domain set is {1,2,3,4} Where as the image set is {1,2,3}.Because f(a)=1 as 1 is the image of a under the rule 'f'. f(b)=2 as 2 is the image of b under the rule 'f'. f(c)=3 as 3 is the image of c under the rule 'f'. f(d)=1as 1 is the image of d under the rule 'f'. Because "image set contains only those elements which are the images of elements found in set X". Note it that here f is one -one but not onto, because there is one element '4' left which is the image of nothing element under the rule 'f'

Antisymmetric relation

Antisymmetric: A relation R on a set A is said to be antisymmetric if (b, a) e R and (a, b) e R then a=b for all a, b A. Or a relation is antisymmetric if and only if there are no pairs of distinct elements a and b with a related to b and b related to a. Be careful student, there is a condition if first statement holds then second must be held. i.e. If (b, a) e R and (a, b) e R then a=b Or you can say that if any order pair holds first condition but not second then it is not antisymmetric otherwise it is antisymmetric. let A={1,2,3,4} and a given relation is R={(1,1)(1,2)(1,3)(1,4)(2,2)(2,3)(2,4)(3,3)(3,4)(4,4)} then R is Antisymmetric. Since (1, 2) belongs to R but (2, 1) not belongs to R. First statement is not hold. Similarly other pairs are not hold first condition. So it is antisymmetric.

How we calculate percentage on excel sheet?

For percentage follow the following rules 1. Select an empty cell. 2. type = sign 3. enter value 4. type % 5. Press enter. For example =12% 0.12