Hand-in @ lecture #10
2.3.18a
For \(f(x) = -3x + 4\) to be a bijection, it needs to be both one-to-one and onto.
Since \(f(x)\) is a linear function with a negative gradient covering \(\mathbb{R}\) for both domain and codomain, it is strictly decreasing. This means that it is indeed both one-to-one and onto, thus \(f(x) = -3x + 4\) is a bijection from \(\mathbb{R}\) to \(\mathbb{R}\).
2.3.18b
For \(f(x) = -3x^{2} + 7\) to be a bijection, it needs to be both one-to-one and onto.
Since \(f(x)\) is a second order polynomial with a negative second order coefficient, it is strictly increasing until a vertex and then strictly decreasing onwards. This means that it is not one-to-one since the codomain is folded
at the vertex meaning that except at the vertex it is two-to-one, nor is it onto since the domain is \(\mathbb{R}\) but the codomain is a true subset of \(\mathbb{R}\)), thus \(f(x) = -3x^{2} + 7\) is not a bijection from \(\mathbb{R}\) to \(\mathbb{R}\).
A-3.2
The following procedure uses assignment statements to interchange the values of the variables x and y:
A minimum of 3 assignment statements are needed to do this, 2 for target assignments and 1 intermediate assignment to hold a value initially held by a variable which would otherwise be overwritten.
3.1.6
The following pseudocode describes an algorithm that takes as input a list of integers and finds the number of negative integers in the list:
3.1.24
The following pseudocode describes an algorithm that determines whether a function from a finite set to another finite set is one-to-one: