Duduli

A function f:X→Y is said to be an onto or surjective function if rng   f =f( X)=Y i.e if every element of Y is the image of some element of X. e.g f:Z→Z,  y=f(x)=x+2 is an onto function Because for every y belongs to Z set, x  belongs to Z set also so the given function is an onto function.

A linear map T: U→V is said to be non singular if it is one-one and onto. Such a map is also called an isomorphism.

The hexadecimal number system consist of 16 digits. These digits are 0,1,2,3 ,4,5,6, 7,8,9,A,B,C,D,E,F. Therefore base of the system is 16.

It is a combinational logic circuit. It is used for subtracting 3 single bit numbers. It has 3 inputs(X,Y & B(in)=Borrow in) from the previous circuit & 2 outputs.[d=difference & B(out )(Borrow output).

A de multiplexer is a combinational logic circuit which receives a single input & distributes it over several output line. Hence a de multiplexer is also known as a distributor since it transmits the signal data to different destination. It is the reverse operation of multiplexers.

Following are the limitations of Rutherford atom model (i) According to electromagnetic theory a revolving electron should continuously emit energy and hence the radius of its path should go on decreasing and ultimately,it should fall into the nucleus. However, electrons revolve around the nucleus without falling in it. Hence, Rutherford atom model cannot explain the stability of atom. (ii) Rutherford model cannot explain line spectra of atoms like hydrogen. Because if the Rutherford atom model is true, the electrons can revolve in orbits of all possible radii hence, it should emit continuous energy spectrum.

A full adder adds two binary bits plus a carry input (Cin) to produce the sum ‘S’ & carry output(C). Block Diagram

A monolithic ic is one in which all circuit components & their interconnections are formed on a single thin wafer called the substrate. Monolithic ics are by far the most common type used in practice. Uses of monolithic ic:- Monolithic ics are used in ★Tv circuits ★computer circuits ★Amplifiers ★voltage regulator , etc.

Half subtractor is a combinational logic circuit. It is used for the purpose of subtracting two signal bit numbers. It has two inputs(A & B) & two outputs  difference (D) & borrow (b). Boolean expressions are D=AB'+A'B    (i.e D is the XOR of A & B) b=A'B.   (i.e b is the AND of A' & B)

Following are merits of shell model : (1) magic numbers are explained. (2) The closed shell structure in shell model explains extra stability and binding energy. (3) The shell model explains the angular momentum of nuclei. (4) The shell model explains nuclear magnetic moments. (5) shell model explains nuclear isomerism.

It states that “the complement of the wholesum is equal to the product of individual complements & vice versa. I.e  (A+B)' = A'.B'[de Morgan's 1st theorem] Similarly    (A.B)' = A'+B' [de Morgan's 2nd theorem]

The velocity with which slowly varying envelope of modulated wave(wave packet) due to group of waves travels through a medium is called group velocity.

Parity of a nucleus refers to the behaviour of the wave function(¥) under inversion of coordinates. e.g if we replace x by -x, y by -y and z by -z and wave function remains the same, then it has even parity. On other hand, if it appears with negative sign, it has odd parity as shown below       ¥(x,y,z)=¥(-x,-y,-z)   even parity       ¥(x,y,z)= - ¥(-x,-y,-z)   Odd parity 

The first experimental evidence of de-Broglie hypothesis came from the experiment of davisson & germer in 1927. It is a matter of common experience that light exhibits diffraction occurance of various maximas can be explained by bragg's law. Davisson & germer showed experimentally that a beam of fast moving electron can also exhibit diffraction. Various maximas can be explained by bragg's law.

As transition of electron takes place from higher orbit to lower orbit, energy will be radiated in the form of radiation. The wave no. depends upon the initial & final orbit. Accordingly no. of series will be obtained (namely lymen,Balmer,paschen, bracket,p-fund series). Each series is composed of no. of lines.

Median is the middle value for a set of data that has been arranged in order of smallest to largest. To find median of n no. of values first we have to arrange the given values in ascending (increasing) order. Median= the value in (n+1/2)th place , if n is odd Median= [ the value in (n/2)th place+the value in (n/2+1)th place ] / 2 , if n is even 

Mode is the most frequent value in a data set. There can be no mode, one mode or multiple modes in a data set. e.g   4,2,8,9,2,7,2,11 here mode=2, since it occurs 3 times in the given data set, which is more than any other value.

As the electron beam leaves the accelerating anode, it comes under the influence of vertical horizontal deflection plates. If no voltage is applied to the deflection plates, the electron beam will appear at O. Where the positive plate electrons are deflected towards that plate.

An octal number can be expressed with the help of digits 0,1,2,......,7. These digits are possible reminders on division by 8. The base or radix of this system is 8.

A karnaugh map (k map) is a visual method(or pictorial method) used to simplify the algebraic expressions in Boolean functions without having to resort to complex theorems or equation manipulation.

A function f:A→B is said to be a constant function if f(x)=C, for every x belongs to A where C is a constant. e.g f:R→R given by f(x)=5 for every x belongs to R is a constant function.

The operation which combines any two elements of a set to get another element of the set is called a binary operation or binary composition on that set. e.g Addition is a binary operation on the set N of natural numbers. Multiplication is a binary operation on the set N of natural numbers.

If Z=a is a singularity of  f(z) & if there is no other singularity within a small circle surrounding the point z=a, then z=a is said to be an isolated singularity of the function f(z)  otherwise it is called non isolated singularity.

The output of xor gate is stated ‘1’ only when it's two input are in different state with respect to each other. It performs a Xor logical operation. xor gate also known as exclusive OR gate. Truth table for xor gate

Set S be a non-empty subset of a vector space V(F). The set of all linear combinations of finite sets of elements of S, is called the linear span of S & is denoted by L(S). Mathematically, L(S)={a1x1+a2x2+...........+anxn : xn belongs to S & ai belongs to F,   i=1,2,3....n}

The number of vectors presents in a basis of a vector space V is called the dimension of V . It is denoted by dim(V). dim(Vn)=n, since there are n number of vectors in a basis of Vn. Here, we are mainly concern with finite dimensional vector space. The dimension of vector space may be infinite.

A series is defined as the sum of the elements of a sequence. If sum of the n-terms of the sequence value (i.e sn) is finite & unique at n→infinity then the given series is a convergent series.        I.e.   lim n→♾️   Sn = finite (unique) Where  Sn = a1+a2+a3+a4+......

Let f be homomorphism (linear transformation) of a vector space U(f) into a vector space V(f). Then the set k of all the elements of U which are mapped into zero element of V is called kernel of linear transformation & is defined as  K = { x1 belongs to U ; f(x1)=0' where 0' is the zero vector of V }

According to this model, nuclei of all the elements behave like a liquid drop. As observed that binding energy of the nucleus is proportional to the number of nucleons. It is similar to binding of molecules in a liquid drop. A small drop is spherical because of surface tension. A nucleus is also assumed to be spherical in shape. The density of a liquid drop is independent of its value. But it depend on the nature of liquid. In case of nucleus, the nuclear density is independent of nuclear volume and it is same for all type of nuclei.

The gamma decay is the spontaneous emission of electromagnetic  Photon from the nucleus. When the nucleus in the excited state passes through lower emission state or ground state from higher energy state then it emits a high energy Photon known as gamma emission.

If F(s) is the laplace transform of a function f(t), then f(t) is known as inverse laplace transform.      L^-¹ {F(s)} = f(t)