The post What is Digital to Analog Converter? appeared first on Goseeko blog.

]]>This DAC uses binary weighted resistors to produce analog output equivalent to the binary input. The binary input is provided to the inverting adder circuit. Below figure represents the weighted resistor DAC. We can represent the binary number in the form of bits b0, b1, b2. The bit b0 is the least significant bit. The bit b1 is the most significant bit. The switches present will jump to ground when the input is equal to zero. When the input is 1 the switches will switch to position -VR which is the negative reference voltage.

The circuit consists of an Op-Amp which is an inverting op-amp as input is provided through the negative terminal. As we know from the virtual ground concept, the value of voltage at inverting and no-invertig end of the op-amp will be the same. Hence, the value of voltage at the input terminal node will be equal to 0V.

The value of current at each node can be determined by writing the nodal equations.

We have a 3-bit binary weighted resistor DAC. As there are only three input bits b0-b2. Then the number of possible outputs can be from 000 to 111 for a particular reference voltage VR.

Now for N-bit binary weighted resistor DAC the general form of output voltage can be

This R-2R ladder DAC consists of R-2R ladder structure as shown below. Then the DAC produces analog output which is equal to the binary input. The ladder network then produces output with an inverting adder circuit in it. The figure below explains the circuit of R-2R ladder DAC.

Similar to Weighted Resistor DAC this circuit also has b0 as the least significant bit and b2 as the most significant bit. When the input bits are equal to 0 the digital switches jump to ground. When the input bits are 1 the switches jump to negative reference value -VR.

We can also easily find out the output voltage equation for this DAC for individual binary inputs. But it is very difficult to get the generalized output voltage equation in this case.

The reason why we prefer R-2R Ladder DAC over Weighted Resistor are

- The design of this type of DAC is easy as only R and 2R resistors are present.
- We can also include more R-2R combinations depending upon the required number of bits.

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]]>The post What is a ring? appeared first on Goseeko blog.

]]>(1) R is an additive Abelian group,

(2) R is an multiplicative semigroup

(3) The two distributive laws hold good, viz

a. (b + c) =a.b + a.c. (left distributive law)

(a + b). c = a.c + b.c, (right distributive law) for all a, b, c R

1. In order to be a ring, there must be a non-empty set equipped with two binary operations, viz, + and . , satisfying the postulates mentioned above. Later on we will simply write ‘‘Let R be a ring’’. It should be borne in mind that there are two binary operations, viz, + and , on R.

2. It should also be borne in mind that the binary operations + and . may not be own usual addition and multiplication.

3. Since R is an additive Abelian group, the additive identity, denoted by 0, belongs to R. We call it the zero element of R. Here 0 is a symbol.

4. If a , bR, then – bR ( R is an additive group). a+ (– b) is, generally, denoted by a – b.

5. a.b is, generally, written as ab.

**Commutative Ring: **R is commutative if ab = ba, for all a, bR.

**R with unity **: R is said to be a ring with unity (or identity) if there exists an element e R such that

ae = a = ea, for all a R.

**Note**: A ring may or may not have a unity.

2. If R is a ring with unity, unity may be same as the zero element. For example, {0} is a commutative ring with unity. Here 0 is the additive as well as the multiplicative identity. It is known as the zero ring.

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]]>The post What are the Increasing and decreasing functions? appeared first on Goseeko blog.

]]>Or in other words, A function is said to be an increasing function in [a, b] if its first derivative is greater than zero for all values of x in an interval [a, b].

And

A function is said to be decreasing function in an interval [a, b] if y decreases as well as x increases from a to b.

“If the first derivative is less than zero for all values of x in an interval [a, b] then the function y = f(x) is a decreasing function in [a, b].

We call a function f(x) is maximum at x = a. If f(a) is greater than every other value of f(x) in the immediate neighbourhood of x = a. Then (i.e., f(x) ceases to increase but begins to increase at x = a.

Similarly, the minimum value of f(x) will be that value at x = b which is less than other values in the immediate neighbourhood of x = b.

As we know that the value of a function at maximum point is the maximum value of a function.

Similarly, the value of a function at minimum point is the minimum value of a function.

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region.

Pierre de Fermat was the first mathematician to discover general method for calculating maxima and minima of a function. The maxima and minima are complement of each other.

If f(x) is a single value function in a region R, then-

Maxima is a maximum point if and only if

Minima is a minimum point if and only if

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]]>The post What is graph theory? appeared first on Goseeko blog.

]]>It also has uses in social sciences, chemical sciences, information retrieval systems, linguistics even in economics also.

Graphs are discrete structures consisting of vertices and edges that connects these vertices.

There are several different types of graphs that differ with respect to the kind and number of edges that can a connect a pair of vertices.

Graph theory is a relatively new area of mathematics. Graph is the form of representing of descriptive data in the terms of verticals and edges. hence, we use Graph theory in various fields like computer science, information technology, genetics, telecommunication etc.

A graph is a collection of vertices and edges in which each edge is assigned to pair of points, we call them terminal.

Hence, we can say that a graph is a network of dots connect by lines.

Mathematically we can define a graph as-

A graph is a pair of set (V, E) where-

1. V is a non-empty set, we call its elements vertices.

2. E is collection of two-element subset of V, we call it edges.

**Terminology: **A graph G is an order pair (V, E) where V is a non-empty set and E is the set of edges in which each element of E is assign to a unique unorder pair of elements of V.

We denote an element of a set E as- e = (v, su) where u, v ∈ V.

hence, U and v are the end vertices of edge e.

**Note- In any graph the number of vertices with odd degree must be even.**

**Loop: **If both the end vertices of an edge are same then we call the edge as a loop.

**Parallel edge: **If two or more edges have same terminal vertices, then we call these edges as parallel edges.

**Interested in learning about similar topics? Here are a few hand-picked blogs for you!**

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]]>The post What are matrices and their types? appeared first on Goseeko blog.

]]>A matrix is a rectangular arrangement of the numbers. We call these numbers inside the matrix as elements of the matrix.

HereThe expression of a matrix ‘A’ is-

The vertical elements are columns and the horizontal elements are rows of the matrix.

The order of matrix A is m by n or (m× n)

We denote a matrix ‘A’ is as-

Where, i = 1, 2, …….,m and j = 1,2,3,…….n

Here ‘i’ denotes row and ‘j’ denotes column.

**1. Rectangular matrix-**

A matrix in which the number of rows is not equal to the number of columns, then we say that the given matrix is a rectangular matrix.

Example:

The order of matrix A is 2×3, hence, it has two rows and three columns.

Matrix A is a rectangular matrix.

**2. Square matrix-**

A matrix with equal number of rows and columns, is a square matrix.

Example:

The order of matrix A is 3 ×3 , that means it has three rows and three columns.

Matrix A is a square matrix.

**3. Row matrix-**

A matrix with a single row and any number of columns is called row matrix.

Example:

**4. Column matrix-**

A matrix with a single column and any number of rows is called row matrix.

Example:

**5. Null matrix (Zero matrix)-**

A matrix in which each element is zero, then we call it a null matrix or zero matrix, therefore, we denote it by O.

Example:

**6. Diagonal matrix-**

A matrix is said to be diagonal matrix if all the elements except principal diagonal are zero

therefore, The diagonal matrix always follows-

So that, for Example:

**7. Scalar matrix-**

A diagonal matrix in which all the diagonal elements are equal to a scalar, hence, we call it a scalar matrix.

Example-

**8. Identity matrix-**

A diagonal matrix is said to be an identity matrix if its each element of diagonal is unity or 1.

Hence, we denote it by – ‘I’

**9. Triangular matrix-**

If every element above or below the leading diagonal of a square matrix is zero. therefore, we call it a triangular matrix.

Hence, there are two types of triangular matrices-

**(a) Lower triangular matrix-**

If all the elements below the leading diagonal of a square matrix are zero, therefore, we call it a lower triangular matrix.

Example:

**(b) Upper triangular matrix-**

If all the elements above the leading diagonal of a square matrix are zero, hence, we call it a lower triangular matrix.

Example –

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]]>The post What is the binary operation? appeared first on Goseeko blog.

]]>**For example:**

Addition + is a binary operation in the set of natural number N, integer Z and real number R.

Multiplication is a binary operation in N, Q, Z, R and C.

**Propery-1: **Let A be any set. A binary operation * A × A →A is commutative if for every a, a, b ∈A.

a* b = b * a

**Property-2:**Let A be a non-empty set. A binary operation *; A × A →A is associative if

a* b) * c = a * (b * c) for every a, b, c ∈A.

**Property-3:** Let * be a binary operation on a non-empty set A. If there exists an element e ∈A such that e * a = a * e = a for every a ∈A, then we call the element e as identity with respect to * in A.

**Property-4:** Let * be a binary operation on a non-empty set A and e be the identity element in A with respect the operation *. An element a ∈A is invertible if there exists an element b ∈A such that

a* b = b * a = e

In which case a andb are inverses of each other. For the operation * if b is the inverses of a ∈A then we can write b =

We denote a bin. operation by * in a set A, is to satisfy.

(i) Left cancellation law if for all a, b, c ∈A,

a* b = a * c ⇒b = c

(ii) Right cancellation law if for all a, b, c ∈A

b* a = c * a ⇒b = c

If A is a set and * is a bin. operation on A, then we call (A, *) an algebraic structure.

**Example: Let R be the set of real numbers, then (R, +) is an algebraic structure.**

**Example: If N denotes the set of natural numbers then (N, +) is an algebraic structure.**

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]]>The post What is the principal of inclusion and Exclusion appeared first on Goseeko blog.

]]>That is, we find the number of elements in A or B (or both) by first adding n(A) and n(B) (inclusion) and then Subtracting n(A ∩B) (exclusion) since its elements were counted twice

Here n(A) is the cardinality of set A, n(B) is the cardinality of set B and n(A⋂B) is the cardinality of (A⋂B).

For three sets-

n(A ∪B ∪C) = n(A) + n(B) + n(C) −n(A ∩B) −n(A ∩C) −n(B ∩C) + n(A ∩B ∩C)

In college, 49 study Physics , 37 study English and 21 study Biology. If 9 n(P⋂E) = 9

n(E⋂B) = 5

n(P⋂B) = 4

n(P⋂E⋂B) = 3

therefore, by using the formula we get

n(P⋃E⋃B) = n(P) + n(E) + n(B) – n(P⋂E) – n(E⋂B) – n(P⋂B) + n(P⋂E⋂B)

= 49 + 37 + 21 – 9 – 5 – 4 + 3

= 92

Hence, the number of students in the college is 92.

of these students study Maths Physics and English, 5 study English and Biology, 4 study Physics and Biology and 3 study Physics, English and Biology, find the number of students in the college.

Solution:

Suppose P represents the number of students who study Physics, E represents the number of students who study English and B represent the number of students who study Biology.

As we know that-

Number of students in the group = n(P⋃E⋃B)

Here it is given that- n(P) = 49, n(E) = 37, n(B) = 21

**Example: Find the number of students at a college choosing at least one of the subject from mathematics, physics and statistics.**

**The data is given as-**

**65 study maths, 45 study physics, 42 study statistics**

**20 study maths and physics, 25 study maths and statistics, 15 study physics and statistics and 8 study all the three subjects.**

Sol.

Here we need to find , where

M maths, P = physics, S = statistics

Hence, by using Inclusion-Exclusion principle-

Which gives-

Therefore, we find that 100 students study at least one of the three subjects.

**Interested in learning about similar topics? Here are a few hand-picked blogs for you!**

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]]>The post What is the present value? appeared first on Goseeko blog.

]]>Or in other words “PV describes how much a future sum of money is worth today”.

Let the objective is to have an amount A after

nyears from today*. *If the interest rate is ‘i’, then the amount is required to deposited now

so as to achieve the set target is-

Which is called the PV of A.

Note-

1. Sum due = PV + Interest on present value

**Present value (Continuous compounding)-**

We use the following formula if there is continuous compounding-

Where-

P = principal amount

A = Amount at future point of time

t = time

r = rate of interest

**Example: Find out the present value of Rs. 1000 due in 2 years at 5% per annum compound interest, If the interest is paid yearly.**

Sol.

Here, A = 1000, n = 2, i = 0.05

We have to find P-

By the formula-

Therefore the PV is Rs.907.03.

**Example: Find the** PV **of Rs. 4,500 due after 3 years from now. the interest is compounded continuously at the interest rate of 6%.**

Sol.

Here we have- A = 4500, t = 3, r = 0.06,

To find P,

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]]>The post What is the principle of mathematical induction? appeared first on Goseeko blog.

]]>An essential property of the set **N **= {1, 2, 3, …} of positive integers follows:

**Principle of Mathematical Induction I: **Let P be a proposition defined on the positive integers **N**; that is, P(n) is either true or false for each n ∈ **N**. Suppose P has the following two properties:

(i) P(1) is true.

(ii) P(k + 1) is true whenever P(k) is true.

Then P is true for every positive integer n ∈ **N**.

We shall not prove this principle. In fact, this principle is usually given as one of the axioms when **N **is developed axiomatically.

**Principle of Mathematical Induction II: **Let P be a proposition defined on the positive integers **N **such that:

(i) P(1) is true.

(ii) P(k) is true whenever P(j) is true for all 1 ≤ j < k.

Then P is true for every positive integer n ∈ **N**.

**Note: **Sometimes one wants to prove that a proposition P is true for the set of integers {a, a + 1, a + 2, a + 3, . . .} where a is any integer, possibly zero. This can be done by simply replacing 1 by a in either of the above Principles of Mathematical Induction.

The technique has two steps to prove any statement-

1. Base step (it proves that a statement is true for the initial value)

2. Inductive step (it proves that is the statement is true for n’th iteration then it is also true for (n+1)’th iteration.

Example:

Sol. here for n = 1, 1 = 1²

Now let us assume that the statement is true for n = k

Hence we assume that is true.

Now we need to prove that-

So that which satisfies the second step.

Hence-

**Example-2:****Prove 1+2+…+n=n(n+1)/2 using a proof by induction**

**Sol.**

Let n=1: 1 = 1(1+1)/2 = 1(2)/2 = 1 is true,

**Step-1: **Assume n=k holds: 1+2+…+k=k(k+1)/2

Show n=k+1 holds: 1+2+…+k+(k+1)=(k+1)((k+1)+1)/2

Substitute k with k+1 in the formula to get these lines. Notice that I write out what I want to prove.

**Step-2: **Now start with the left side of the equation

1+2+…+(k+1)=1+2+…+k+(k+1)

=k(k+1)/2 + (k+1)

=(k(k+1) + 2(k+1))/2 by 2/2=1 and distribution of division over addition

=(k+2)(k+1)/2 by distribution of multiplication over addition

=(k+1)(k+2)/2 by commutativity of multiplication.

**Example-3: Prove the following by using the principle of mathematical induction for all n ****∈**** N-**

Sol. Here, n = 1, which is true

Step-1: Assume n = k holds-

Now show n = k + 1 also holds-

Consider-

Which is also true for n = k + 1.

Hence proved.

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]]>The post What are Logical connectives? appeared first on Goseeko blog.

]]>p∧q

Read “*p *and *q*,” denotes the conjunction of *p *and *q*. Since *p *∧*q *is a proposition it has a truth value, and this truth

Value depends only on the truth values of *p *and *q*.

Note- If *p *and *q *are true, then *p *∧*q *is true; otherwise *p *∧*q *is false.

Any two propositions can be combined by the word “or” to form a compound proposition called the disjunction of the original propositions. Symbolically,

*p*∨*q*

Read “*p *or *q*,” denotes the disjunction of *p *and *q*. The truth value of *p *∨*q *depends only on the truth values of *p*

And *q *as follows-

If p and q are false, then p ∨q is false; otherwise p ∨q is true

Given any proposition p, another proposition, called the negation of p, can be formed by writing “It is not true that . . .” or “It is false that . . .” before p or, if possible, by inserting in p the word “not.” Symbolically, the negation of p, read “not p,” is denoted by

￢p

The truth value of ￢p depends on the truth value of p as follows-

If p is true, then ￢p is false; and if p is false, then ￢p is true

**Conjunction- p ⋀ q**

Any two propositions can be combined by the word “and” to form a compound proposition called the conjunctionof the original propositions. Symbolically,

p∧q

Read “*p *and *q*,” denotes the conjunction of *p *and *q*. Since *p *∧*q *is a proposition it has a truth value, and this truth

Value depends only on the truth values of *p *and *q*.

Note- If *p *and *q *are true, then *p *∧*q *is true; otherwise *p *∧*q *is false.

**2. Disjunction, p ****∨****q**

Any two propositions can be combined by the word “or” to form a compound proposition called the disjunctionof the original propositions. Symbolically,

*p*∨*q*

Read “*p *or *q*,” denotes the disjunction of *p *and *q*. The truth value of *p *∨*q *depends only on the truth values of *p*

And *q *as follows-

If p and q are false, then p ∨q is false; otherwise p ∨q is true

**Negation-**

Given any proposition p, another proposition, called the negation of p, can be formed by writing “It is not true that . . .” or “It is false that . . .” before p or, if possible, by inserting in p the word “not.” Symbolically, the negation of p, read “not p,” is denoted by

￢p

The truth value of ￢p depends on the truth value of p as follows-If p is true, then ￢p is false; and if p is false, then ￢p is true

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]]>