elements

Examples of elements in the following topics:

• Total Number of Subsets

  • The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc.
  • These numbers also arise in combinatorics, where $n^b$ gives the number of different combinations of $b$ elements that can be chosen from an $n$-element set.
  • The total number of subsets is the number of sets with 0 elements, 1 element, 2 elements, etc.
  • The total number of subsets of a set with $n$ elements is $2^n$.
  • It has 6 elements, therefore, $2^n=2^6=64$ subsets.

• Permutations of Nondistinguishable Objects

  • But what if some elements are repeated?
  • Repetition of some elements complicates the calculation of permutations, because it allows for there to be multiple ways in which a specific order of elements can be arranged.
  • To correct for the "multiplicity" of certain permutations, we must divide the factorial of the total number of elements by the product of the factorials of the number of each repeated element.
  • $ distinct ways, $3$ elements can be arranged in a total of $3!
  • In total, there are $13$ elements.

• One-to-One Functions

  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain.
  • In other words, every element of the function's range corresponds to exactly one element of its domain.
  • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
  • The ordered pairs $(-2,4)$ and $(2,4)$ do not pass the definition of one-to-one because the element $4$ of the range corresponds to to $-2$ and $2$.

• Permutations of Distinguishable Objects

  • The number of permutations of distinct elements can be calculated when not all elements from a given set are used.
  • If not all of the objects in a set of unique elements are chosen, the following formula is used.
  • This formula determines the number of possible permutations of $k$ elements selected from the set of $n$ elements:
  • To solve this problem, we want to evaluate the number of possible permutations of  $3$elements from the set of $25$ elements; in other words, $k = 3$ and $n=25$.
  • In other words, the order of the elements selected does matter.

• Matrix Multiplication

  • When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.
  • Scalar multiplication is simply multiplying a value through all the elements of a matrix, whereas matrix multiplication is multiplying every element of each row of the first matrix times every element of each column in the second matrix.  
  • When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.
  • Start with producing the product for the first row, first column element.  
  • Take the first row of Matrix $A$ and multiply by the first column of Matrix $B$:  The first element of $A$ times the first column element of $B$, plus the second element of $A$ times the second column element of $B$.

• What is a Matrix?

  • The individual items (numbers, symbols or expressions) in a matrix are called its elements or entries.  
  • Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element.
  • Any matrix can be multiplied element-wise by a scalar from its associated field.
  • Each element of a matrix is often denoted by a variable with two subscripts.
  • For instance, $a_{2,1}$ represents the element at the second row and first column of a matrix A.

• Addition and Subtraction; Scalar Multiplication

  • Just add each element in the first matrix to the corresponding element in the second matrix.
  • Note that element  in the first matrix, $1$, adds to element $x_{11}$ in the second matrix, $10$, to produce element $x_{11}$ in the resultant matrix, $11$.
  • Multiplying a matrix by $3$ means the same thing; you add the matrix to itself $3$ times, or simply multiply each element by that constant.

• Sequences of Mathematical Statements

  • In mathematics, a sequence is an ordered list of objects, or elements.
  • The length of a sequence is the number of ordered elements, and is possibly infinite.
  • Unlike a set, order matters in sequences and exactly the same elements can appear multiple times at different positions in the sequence.

• Further Simplification of Matrices

  • Divide the diagonal element and the right-hand element (of b) for that diagonal element's row so that the diagonal element is equal to one.
  • Divide each element of the diagonal and the corresponding row element of b by a number which yields a 1 in the diagonal:

• Sets of Numbers

  • Every element of a set must be unique; no two members may be identical.
  • The order in which the elements of a set are listed is irrelevant (unlike for a sequence).
  • because the extensional specification means merely that each of the elements listed is a member of the set.
  • For sets with many elements, the enumeration of members can be abbreviated.
  • A subset is a set whose every element is also contained in another set.