MTH100 Definitions by VUBuddy

Q1. Define a set?

Definition: A set is an unordered collection of distinct objects. Objects in the collection are called elements of the set.

Q2. What is roster method of sets?

The roster method of specifying a set consists of surrounding the collection of elements with braces. For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.

Q3. Define null set?

Definition: The set with no elements is called the empty set or the null set.

Q4. Define universal set?

Definition: The universal set is the set of all things pertinent to a given discussion and is designated by the symbol U

Q5. Define subset?

Definition: The set A is a subset of the set B, denoted A  B, if every element of A is an element of B.

Q6. Define equal set?

Definition: Two sets A and B are equal if A  B and B  A. If two sets A and B are equal we write A = B to designate that relationship.

Q7. Define intersection of sets?

Definition: The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B. We write A  B to denote A Intersection B. Example: If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A  B = {3, 6}

Q8. Define union of sets?

Definition: The union of two sets A and B is the set containing those elements which are elements of A or elements of B. We write A  B to denote A Union B. Example: If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A  B = {1, 2, 3, 4, 5, 6}.

Q9. What are algebraic properties of sets? Or difference between commutative assosiative and distributive law?

Commutative: Union and intersection are commutative operations. In other words, A

B = B

 A and A ∩ B = B ∩ A

Assosiative: Union and intersection are associative operations. In other words, (A

B) ·  C = A ·  (B ·  C) and (A ∩ B) ∩ C = B ∩ (A ∩ C)

Distributive: Union and Intersection are distributive with respect to each other. In other words A ∩ ( B

 C )= (A ∩ B)

 (A ∩ C) and A

 ( B ∩ C )= (A ·  B) ∩ (A ·  C)

Q10. Define cardinality? With two types?

Definition: Cardinality refers to the number of elements in a set A finite set has a countable number of elements An infinite set has at least as many elements as the set of natural numbers

Q11. Define complex number?

Definition: Numbers of the form a + bi are called complex numbers. a is the real part and b is the imaginary part. The set of complex numbers is denoted by C

Q12. Defineabsolute value?

Definition: The absolute value or modulus of a complex number is the distance the complex number is from the origin on the complex plane.

Q13. Define relation?

Definition: A mapping between two sets A and B is simply a rule for relating elements of one set to the other. A mapping is also called a relation.

Q14. Define domain and range ?

Definition: The set consisting of members of the pre-image or inputs of a function is called its domain. For a given domain the set of possible outcomes or images of a function is called its range.

Q15. Define even and odd function?

Definition: A function is called an even function if its graph is symmetric with respect to the vertical axis, and it is called an odd function if its graph is symmetric with respect to the origin.

Q16. Define quadratic function?

Definition: A function of the type y = ax 2 + bx + c where a, b, and c are called the coefficients, is called a quadratic function.

Q17. Define a matrix?

Definition: A matrix is a rectangular arrangement of numbers in rows and columns. The order of a matrix is the number of the rows and columns. The entries are the numbers in the matrix.

Q18. Define identity matrix?

Definition: A Square matrix with ones on the diagonal and zeros elsewhere is called an identity matrix. It is denoted by I.

Q19. When a matrix has echelon form?

A matrix is in echelon form if it has the following properties Every non-zero row begins with a 1 (called a leading 1) Every leading one in a lower row is further to the right of the leading one above it. If there are zero rows, they are at the end of the matrix

Q20. When a matrix has refuced echelon form?

A matrix is in reduced echelon form if in addition to the above three properties it also has the following property: Every other entry in a column containing a leading one is zero Methods for finding Solutions of Equations: Using Row Operations: Recall that when we are solving simultaneous equations, the system of equations remains unchanged if we perform the following operations: Multiply an equation by a non-zero constants Add a multiple of one equation to another equation Interchange two equations.

Q21. Define adjoin or adjugate?

Definition: Given a matrix A, calculate all the cofactors of A. We then form the matrix ( of the cofactors. The Adjoint or Adjugate of A is the transpose of the matrix of the cofactors.

Q22. Define sequence and terms?

Definition: Rows of numbers are called sequences, and the separate numbers are called terms of the sequence.

Q23. Define arithmetic sequences?

Definition: An Arithmetic Sequence (or Arithmetic Progression) is a sequence in which each term after the first term is found by adding a constant, called the common difference (d), to the previous term

Q24. Define geometric sequences?

Definition: A sequence in which each term after the first is found by multiplying the previous term by a constant value called the common ratio, is called a Geometric Sequence (or Geometric Progression). The formula for finding any term of a geometric sequence is an = ar n-

Q25. Define geometrics series?

Definition: A Geometric Series is the sum of the terms in a arithmetic sequence. The formula for fining the sum of the first n terms of a geometric sequence is given by Sn=a(1-rn)/1-

Q26. Define convergent and divergent?

Definition: If a sequence of numbers approaches (or converges) to a finite number, we say that the sequence is convergent. If a sequence does not converge to a finite number it is called divergent.

Q27. What is multiplication principal?

Multiplication Principle: If two operations A and B are performed in order, with n possible outcomes for A and possible outcomes for B, then there are n x m possible combined outcomes of the first operation followed by the second.

Q28. Describe Pascal’s Triangle?

Expression= Coefficients

(x + y) 1 = x + y= 1 1

(x + y) 2 = x 2 + 2xy + y=

(x + y) 3 = x 3 + 3x 2y + 3xy 2 + y 3= 1 3 3 1

(x + y) 4 = x 4 + 4x 3y + 6x 2y 2 + 4xy 3 + y=

(x + y) 5 = x 5 + 5x 4y + 10x 3y 2 + 10x 2y 3 + 5xy 4 + y 5 =1 5 10 10 5 1

Q29. What are Properties of Gradient?

The bigger the gradient’s magnitude is, the steeper the line segment. Negative gradient means line is facing downwards. Positive gradient means the line is facing upwards. The slope gives the average rate of change in y per unit change in x, where the value of y depends on x. Two line segments that are parallel will have the same slope.

Q30. What are Properties of the Cosine Function?

The cosine function has “period” 360 o as it repeats itself after each revolution of 360 o Cos(-  ) = Cos  as the x-coordinate of P doesn’t change when we reflect across the xaxis. Cos (180 -  ) = - Cos  as the x-coordinate changes signs when reflected across the yaxis. Cos (  - 180) = - Cos  as the x-coordinate changes signs when reflected across the origin. Cos  is positive in the first and the fourth quadrant (as the x coordinate of P is positive), and negative in the second and the third quadrant as the x-axis is negative there. The range of the cosine function is between -1 and 1. The maximum value of 1 is taken when  = 0 o, ±360 o, ±720 o, …, and the minimum value of -1 is at  = ±180 o, ±540 o, …,

Q31. Define periodicfunction and period?

The functions with the property that they keep repeating themselves are called periodic functions. The smallest interval for which the function repeats itself is called its period.

Q32. What are Properties of the Sine Function?

Sin(-  ) = -Sin  as the y-coordinate of P changes sign when we reflect across the x-axis. Sin  is positive in the first and the second quadrant (as the y coordinate of P is positive), and negative in the third and the fourth quadrant as the y coordinate is negative there. Sin(180 -  ) = Sin  because as we reflect across the y-axis the y-coordinate doesn’t change. Sin(  - 180) = - Sin  as the y-coordinate changes signs when reflected across the origin. Like the cosine function, the sine function is also periodic, with period 360 degrees, and range between -1 and 1.