D420 Discrete Math: Functions and Relations - Set 3 - Part 2

Correct Answer: If a is related to b and b is related to a, then a=b

An antisymmetric relation is one where if a is related to b and b is related to a, then a must equal b.

Correct Answer: If a is related to b and b is related to c, then a is related to c

A transitive relation is one where if a is related to b and b is related to c, then a is also related to c.

Correct Answer: It is reflexive, symmetric, and transitive

An equivalence relation is one that satisfies the properties of being reflexive, symmetric, and transitive.

Correct Answer: A binary relation where every pair of elements is comparable

A total order is a binary relation where every pair of elements is comparable, meaning one element is always either less than or greater than the other.

Correct Answer: A binary relation that is transitive and antisymmetric

A strict order is a binary relation that is transitive and antisymmetric, meaning no element is related to itself, and the relation is preserved through transitivity.

Correct Answer: To represent the order of elements in a partially ordered set

A Hasse diagram is a graphical representation of a partially ordered set, showing the order relations between the elements.

Correct Answer: It contains no cycles

A directed acyclic graph (DAG) is a graph with directed edges and no cycles, meaning there is no way to return to the starting vertex by following the edges.

Correct Answer: A square matrix that has an inverse

An invertible matrix is a square matrix that has an inverse, meaning it can be multiplied by another matrix to yield the identity matrix.

Correct Answer: A method for solving systems of linear equations by reducing a matrix to row echelon form

Gaussian elimination is a method used to solve systems of linear equations by reducing the coefficient matrix to row echelon form.

Correct Answer: The number of non-zero rows in its row echelon form

The rank of a matrix is the number of non-zero rows in its row echelon form, which corresponds to the dimension of the row space.

Correct Answer: A matrix that includes an extra column representing the constants from a system of equations

An augmented matrix is used in solving systems of linear equations and includes an extra column for the constants from the equations.

Correct Answer: All of the above

Elementary row operations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another.

Correct Answer: A scalar value that can indicate whether a matrix is invertible

The determinant of a square matrix is a scalar value that can be used to determine whether the matrix is invertible. If the determinant is zero, the matrix is not invertible.

Correct Answer: Swapping the rows and columns of a matrix

Transposing a matrix means swapping its rows and columns, resulting in a new matrix where the element at position (i,j) in the original matrix becomes the element at (j,i).

Correct Answer: The product of the matrix and its inverse is the identity matrix

The inverse of a square matrix is such that when the matrix is multiplied by its inverse, the result is the identity matrix.

Correct Answer: It has 1s on the diagonal and 0s elsewhere

The identity matrix is a square matrix with 1s on the diagonal and 0s in all other positions.

Correct Answer: The sum of the diagonal elements of the matrix

The trace of a square matrix is the sum of its diagonal elements.

Correct Answer: A matrix where all non-diagonal elements are zero

A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.

Correct Answer: The transpose of the transpose is the original matrix

Transposing a matrix twice returns the original matrix.

Correct Answer: Reducing a matrix to row echelon form

Gaussian elimination is a process used to reduce a matrix to row echelon form, simplifying the process of solving systems of linear equations.