Set Theory’s easy and fundamental concepts, such as Union, Intersection, and , are used extensively in Algebra, Logic, and Probability. As a result, this subject is extremely important in banking and MBA exams.
Determinants aid in determining the adjoint or inverse of a matrix. This notion is also required to solve linear equations using the matrix inversion approach. The cross-product of two vectors is easily recalled by calculating determinants. Due to these reasons, one must know the and how to solve them to score well in linear algebra.
On that note, let’s discuss all the basics of the topics in length from definition and types to properties and operations to help you ace any Sets/Determinants questions asked in any exam.
Sets
Sets are collections of well-defined items or elements that do not vary from one person to the next. A group is denoted by a capital letter.
They can be in the form of a set builder or a roster. Curly brackets are commonly used to represent sets; for example, A = {1,2,3,4} is a set.
Types of Sets
There are several types of sets in mathematics. Empty sets, finite and infinite sets, equal sets, disjoint sets, subsets, supersets, power sets, universal sets, and so on are examples.
Operations on Sets
The basic operations on sets are:
Union of Sets
If two sets A and B exist, a union B is the set that contains all of the elements from both sets. It is written A∪B.
For instance, if A = {1,3,5} and B = {2,4,6} then A union B is:
A ∪ B = {1,2,3,4,5,6}
Intersection of Sets
If A and B are two sets, then set A intersection B includes only the elements that both sets share. It is written A∩B.
If A = {1,3,5} and B = {2,4,6} the A intersection B is:
or A∩B = { } or Ø
Because A and B share no elements, their intersection yields a null set.
Complement of Sets
The complement of any set, such as P, is the collection of all universal set elements that do not belong to set P. It is represented by the letter P.
Cartesian Product of Sets
If A and B are two sets, then their cartesian product is a set that contains all ordered pairs (a,b) where an is an element of A and b is an element of B. It is denoted by A.B.
For instance: set A = {1,3,5} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(3,Bat),(3,Ball),(5,Bat),(5,Ball)}
Determinants
Determinants are scalar numbers calculated by adding the products of the elements of a square matrix and their cofactors according to a predefined method.
Some Fundamental Properties of Determinants
Reflection Property
The determinant remains unaltered if the rows of the determinant are translated into columns and the columns into rows. This is known as the property of reflection.
All-zero Property
The determinant is zero if all of the elements in a row (or column) are zero.
Switching Property
The swap of any two rows changes the sign of the determinant (or columns).
Scalar Multiple Property
If all of the elements of the determinant’s row (or column) are multiplied by a non-zero constant, the determinant is multiplied by the same constant.
Sum Property
The determinant can be described as a sum of two or more determinants if a few rows or columns are expressed as a sum of terms.
Property of Invariance
If each element of a determinant’s row and column is multiplied by the equimultiples of the elements of another determinant’s row or column, the determinant’s value remains constant.
Triangular Property
Assume the items above and below the major diagonal are both zero. In that case, the value of the determinant is equal to the product of the diagonal matrix’s elements.
Factor Property
A determinant Δ becomes zero when we set it to zero.
If we assume that x=a, then (x-a) is a factor of Δ.