Chapter 2: Q22E (page 184)
Let A be a matrix. Show that the matrix is symmetric. [Hint: Show that this matrix equals its transpose with the help of Exercise 17b.]
Therefore, it is shown that is symmetric.
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Let and let for all . Show that f(x) is strictly increasing if and only if the functionrole="math" localid="1668414567143" is strictly decreasing.
Question: Consider these functions from the set of students in a discrete mathematics class. Under what conditions is the function one-to-one if it assigns to a student his or her
Mobile phone number
Student identification number
Final grade in the class
Home town.
a) Define what it means for a function from the set of positive integers
to the set of positive integers to be one-to-one
b) Define what it means for a function from the set of positive integers to the set
of positive integers to be onto.
c) Give an example of a function from the set of positive integers to the set of
positive integers that is both one-to-one and onto.
d) Give an example of a function from the set of positive integers to the set of
positive integers that is one-to-one but not onto.
e) Give an example of a function from the set of positive integers to the set of
positive integers that is not one-to-one but is onto.
f) Give an example of a function from the set of positive integers to the set of
positive integers that is neither one-to-one nor onto.
a) Prove that a strictly decreasing function from R to itself is one-to-one.
b) Give an example of a decreasing function from R to itself is not one-to-one.
Question: Let\(f(x) = ax + b\) and \(g(x) = cx + d\) where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that \(f \circ g = g \circ f\)
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