Sets

In this I will blog I will solve some questions on the topics I learned in my previous blog.

1. What are sets?

https://wtfuniverse.wordpress.com/2019/04/22/what-are-sets/

2. What are different types of sets?

https://wtfuniverse.wordpress.com/2019/04/22/what-are-different-types-of-sets/

3. What are the different types of operations on sets?

https://wtfuniverse.wordpress.com/2019/04/22/what-are-the-different-types-of-operations-on-sets/

4. What are the different laws of alegbra of sets?

https://wtfuniverse.wordpress.com/2019/04/22/what-are-the-different-types-of-operations-on-sets/

Now to the questions:

Q1. For any setA, (A’)’ is equal to

(a) A’ (b) A (c) Φ (d) none of these.

A1. (b) because the complement of a complement is the original set.

Q2. LetA and B be two sets in the same universal set. Then, A -B =

(a) A∩B (b) A’∩B (c) A∩B’ (d) none of these.

A2. (c) because the set contains all the elements of A-B.

Q3. The number of subsets of a set containing n elements is

(a) n (b) 2ⁿ- 1 (c) n² (d) 2ⁿ

A3. (d) read my blog no 2.

Q4. For any two sets A and B, A ∩ (A ∪ B) =

(a) A (b) B (c) Φ (d) none of these

A4. (a) draw a Venn diagram if u were not able to solve it.

Q5.Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that n(A)=200,n(B)=300 and n(A∩B)=100 .Then,n(A’∩B’)=

A5.

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What are different laws of alegbra of sets?

Hey guys, here with some more math topics. If A, Band C are some sets,then

1. Idempotent law:

A∪A=A

A∩A=A

2. Identity law

A∪φ=A

A∩U=A

3. Associative law

A∪(B∪C)=(A∪B)∪C

A∩(B∩C)=(A∩B)∩C

4. Commutative law

A∪B=B∪A

A∩B=B∩A

5. Distributive law

A∪(B∩C)=(A∪B)∩(A∪C)

A∩(B∪C)=(A∩B)∪(A∩C)

6. De-Morgans law

(A∪B)′=A′∩B′

(A∩B)′=A′∪B′

I have mentioned some of the important laws of algebra of sets, if u have some problem you can understand the laws by drawing there Venn diagram.

Now I would like to mention some important results of sets that you want to remember while doing set problems.

If A, B, C are sets and n(A) represent the number of elements in set A, then

1. n(A∪B)=n(A) + n(B) – n(A∩B)

2. n(A∩B)=n(A) – n(A-B)

3. n(A∪B∪C) =n(A)+n(B)+n(C)-n(A∩B) – n(B∩C)-n(A∩C) +n(A∩B∩C)

4. n(A∪B)′=n(U) – n(A∪B)

5. n(A∩B)′=n(U) – n(A∩B)

What are the different types of operations on sets?

1. Union :

Union of two sets is a set which contain elements of both sets. The operation is denoted by the sign “∪”.

Suppose we have two sets A and B then there union will be represented as A ∪ B. The blue colored region represents A ∪ B. U represent the universal set.

Example

If A = {x:x = 2n +1,n ∈ Z} and B={x:x = 2n,n ∈ Z}, then A∪B =

(a)N (b)Z (c)R

Ans. (b) since A∪B contain both values 2n and 2n+1 hence it contains all Z values.

2. Intersection :

Intersection of two sets are the set of elements common between the sets. The operation is represented by the sign “∩”.

Suppose we have two sets A and B then, the intersection set will be represented as A∩B. The green colored part represents A∩B.

Example

If A = {x:x = 4n, n ∈ Z} and B={x:x = 6n, n ∈ Z}, then A∩B =

(a) {x:x =2n, n ∈ Z}

(b) {x:x = 10n, n ∈ Z}

(c) {x:x = 12n, n ∈ Z}

(d) {x:n = 24n, n ∈ Z}

Ans. (c), A∩B will contain values which are multiple of both 6 and 4, so it will contain the multiple of 12 that why (c) is correct.

3. Difference :

Suppose there are two sets A and B then, the difference of these sets will be represented as”A – B” and will contain values which belongs to only A and not B. The red colored part represents A-B.

4. Symmetric difference :

Suppose we have two sets A and B then, symmetrical difference of these sets is (A-B)∪(B-A), the symmetric difference is represented by A∆B. The purple region represents A∆B.

A∆B=(A∪B) – (A∩B)

5. Compliment of a set :

Let A be a set and U be the universal set then A complement is U-A and its represented by A’.

A’= U-A. The white color represents A’ and yellow represents A.

What are different types of sets?

1. Empty sets:

The sets which contain no elements are called empty sets, these are represented by {} or Φ.

Example

Which of these is the empty set

(a) {x : x is a real number and x² – 1=0}

(b) {x : x is a real number and x² +1=0}

(c) {x : x is a real number and x² – 9=0}

(d){x : x is a real number and x²=x+2}

Ans (b) is the empty set because x=√(-1), sqrt of – 1 is not a real number.

2.Singleton set:

It is a set which has only one element.

Eg

{f}

3. Finite set:

It is a set whose elements can be counted by natural numbers 1,2,3…… and gets terminated at a certain natural number.

For example the set of odd natural numbers less than 10

{1,3,5,7,9}

4. Subset :

If a set A contains all the elements of set B then B is said to be the subset of A, it is represented as B ⊂ A.

Every set is a subset of itself.

If a set contains n elements then the number of its subsets are 2ⁿ.

5. Power set :

The set of all subsets of a set is called its power set.

A={1,2}

P(A)={ {1},{2},{1,2},Φ}

6.Equal sets

The sets which are subsets of each other are called equal sets.

Example

A={1,2}

B={2,1}

A and B are equal subsets.

7. Universal sets :

The set containing all the sets in a given context is called universal sets.

What are sets?

Sets are a collection of well defined objects. Like a collection of all prime number, even numbers, odd numbers etc.

Example:

Which of the following collection is a set?

(a) the collection of all girls in your class.

(b) the collection of intelligent girls in your class.

(c) the collection of beautiful girls in your class.

(d) the collection of tall girls in your class.

Ans. (a) the correct aption because all the other properties in options (b), (c), (d) like intelligence, beautifulness and tallnes are vaguely defined and cannot be used to make a well defined set.👍👍👍👍😳

Sets are usually represented by capital letters A, B, C……….. X, Y, Z.

Suppose ‘a’ is an element of set B then we represent it as a ∈ B which is read as “a belongs to B”. If it doesn’t belong to B then it is represented by a B.

Sets can be represented by two methods :

1. Roster method :

In this method we write all the elements of the set in between “{}” and objects are separated by comma “,” .

Suppose we have to write the set of all vowels, so the set will be represented as {a, i, o, e, u}.

2. Set builder method :

In this method the set is represented by a property of the set, say a function f(x) which holds for x, then the set of x can be represented as {x:f(x)holds}, this is read as “the set of x where f(x) holds” , this can be better explained by an example

Suppose we have the square of all natural numbers then it’s is represented as

{x²:x∈natural numbers}

This can be written in rooster form as

{1,4,9,16,15………}

Example

If B is the set whose elements are obtained by adding 1 to each of the even numbers, then the set builder notation of B is

(a) B = {x : x is even}

(b) B = {x : x is odd and x>1}

(c) B = {x : x is odd and x ∈ Z}

(d) B = {x : x is an integer}

Ans.(c) is the correct answer because whenever we add 1 to an even number we get an odd number.

In (a) we get a set of even numbers which is wrong

In (b) we get a set of odd numbers but we don’t get negative values.

In (d) we get both even and odd values which is wrong. 👍👍👍👍😳

Physics And Mathematics

Now I want to solve some questions based on the concepts introduced in my

blogs :1. What are the Differences between vector and scalar quantities?

https://wtfuniverse.wordpress.com/2019/03/26/what-are-the-difference-between-scalar-and-vector-quantities/

2. How to add vectors?

https://wtfuniverse.wordpress.com/2019/03/26/how-to-add-vectors/

3. What happens when we multiply an integer with a vector?

https://wtfuniverse.wordpress.com/2019/03/26/what-happens-when-we-multiply-a-integer-with-a-vector/

4. How to subtract vectors?

https://wtfuniverse.wordpress.com/2019/03/26/how-to-subtract-vectors/

5. How does a vector behave?

https://wtfuniverse.wordpress.com/2019/03/28/how-does-a-vector-behave/

6. How to do cross product of two vectors?

https://wtfuniverse.wordpress.com/2019/03/28/how-to-do-cross-product-of-two-vectors/

7. How to find rate of change of quantity?

https://wtfuniverse.wordpress.com/2019/03/28/how-to-find-rate-of-change-of-a-quantity/

8. How far will a thing go if we throw it upwards?

https://wtfuniverse.wordpress.com/2019/04/04/how-far-will-a-thing-go-if-we-throw-it-upwards/

9. How to find area of a function?

https://wtfuniverse.wordpress.com/2019/04/08/how-to-find-area-of-a-function/

10. What is the significance of least significant digit?

https://wtfuniverse.wordpress.com/2019/04/08/what-is-the-significance-of-least-significant-digit/

11. What digits are significant in calculation?

https://wtfuniverse.wordpress.com/2019/04/08/what-digits-are-significant-in-calculation/

12. How do physicist remove random errors from experiment?

https://wtfuniverse.wordpress.com/2019/04/08/how-do-physicist-remove-random-errors-in-experiments/

Now to the questions

Q1. Two vectors having equal magnitudes A make an angle θ with each other. Find the magnitude and direction of the resultant.A1.Since α = θ/2 that means the resultant makes an of θ/2 with A vector.Q2. Two vectors of equal magnitude 5 unit have an angle 60 between them. Find the magnitude of (a) the sum of the vectors and (b) the difference of the vectors.A2.Q3. A force of 10.5 N acts on a particle along a direction making an angle of 37 with the verticle. Find the component of the force in the vertical direction.A3.Q4. The work done by a force ⃗F during a displacement ⃗r is given by ⃗F.⃗r. Suppose a force of 12 N acts on a particle in vertically upward direction and the particle is displaced through 2.0m in vertically downward direction. Find the work done by the force during this displacement.A4.Q5. The vector ⃗A has a magnitude of 5 unit, ⃗B has a magnitude of 6 unit and the cross product of ⃗A and ⃗B has a magnitude of 15 unit. Find the angle between ⃗A and ⃗B.A5.Q6. Find dy/dx, if y=eˣsinx.A6.Q8. Evaluate ∫⁶₃ (2x₂ + 3x + 5) dx.A8.Q9. Round off the following numbers to three significant digits(a) 15462(b)14.745(c) 14.750(d) 14.650 * 10¹².A9.Q10. Evaluate (25.2 * 1374)/33.3 . All the digits in this expression are significant.A10.Q11. Evaluate 24.36 + 0.0623 + 256.2.A11.Q12. The focal length of a concave mirror obtained by a student in repeated experiments are given below. Find the average focal length with uncertainty in +_ σ limit.No of obs focal length in cm1 25.42 25.23 25.64 25.15 25.36 25.27 25.58 25.49 25.310 25.7A12.Q13. A vector has component along the x-axis equal to 25 unit and along the y-axis along to 60 unit. Find the magnitude and direction of the vector.A13.Q14. Find the resultant of the three vectors shown in the figure.A14.Q15. The sum of the three vectors shown in figure is zero. Find the magnitudes of the vectors ⃗OB and ⃗⃗OC.A15.Q16. The magnitudes of vectors ⃗OA, ⃗OB and ⃗OC in figure are equal. Find the direction of ⃗OA + ⃗OB – ⃗OC .A16.Q17. Find the resultant of the three vectors ⃗OA, ⃗OB and ⃗OC shown in the figure. Radius of the circle is R.A17.Q18. The resultant of vectors ⃗OA and ⃗OB is perpendicular to ⃗OA. Find the angle AOB.A18.Q19. Write the unit vector in the direction of ⃗A=5 ⃗i+ ⃗j -2 ⃗k.A19.Q20. If |⃗a + ⃗b|=| ⃗a – ⃗b|show that ⃗a and ⃗b are perpendicular.A20.Q21. If ⃗a = 2 ⃗i + 3 ⃗j +4 ⃗k and ⃗b= 4 ⃗i +3 ⃗j+2 ⃗k find the angle between ⃗a and ⃗b.A21.Q22. The volume of a sphere is given by V=(4/3)ΠR³Where R is the radius of the sphere. (a)Find the rate of change of volume with respect to R. (b) Find the change in volume of the sphere as the radius is increased from 20.0 cm to 20.1 cm. Assume that the rate does note appreciably change between R=20.0 cm to R=20.1 cm.A22.Q23. Find the derivative of the following functions with respect to x. (a) y=x²sinx, (b) y=sinx/x and (c) y=sin(x²).A23.Q24. Find the maximum or minimum values of the function y=x+1/x for x>0.A24.Q25. Figure shows the curve y=x². Find the area of the shaded part between x=0 and x=6.A25.Q26. Evaluate ∫ˣ₀ A sinωx dx where A and ω are constants.A26.Q27. The velocity v and displacement x of a particle executing simple harmonic motion are related asvdv/dx = – ω²x.At x=0,v=v₀ . Find the velocity v when the displacement becomes x.A27.Q28. The charge flown through a circuit in the time interval between t and t+dt is given by dq = e^(-t/Τ) dt, where T is a constant. Find the total charge flown through the circuit t=0 to t=T.A28.Q29. Evaluate (21.6002 + 234 + 2732.10)*13A29.

How do physicist remove random errors in experiments?

Random errors other than faulty equipment and carelessness of physicist accur due to random causes thus the name. When the same experiment is done multiple times these errors are sometimes positive and sometimes negative. Thus the average of these results is taken, but the truth of this result is still uncertain. This uncertainty is calculated by calculating standard deviation.

For example :

Suppose x₁, x₂, x₃,….. xₙ are results of a experiment dome many times. So

The best value of x is its mean with a uncertainity of σ.

For eg:

x=5+_ (1.09)