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qr code vb.net open source 1/10i in Visual Studio .NET
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Review Questions and Answers
Part Two
This is not a test! It s a review of important general concepts you learned in the previous nine chapters Read it though slowly and let it sink in If you re confused about anything here, or about anything in the section you ve just finished, go back and study that material some more 11
Question 111 What s a mathematical relation
Answer 111 A relation is a clearly defined way of assigning, or mapping, some or all of the elements of a source set to some or all of the elements of a destination set Suppose that X is the source set for a relation, and Y is the destination set for the same relation In that case, the relation can be expressed as a collection of ordered pairs of the form (x,y), where x is an element of set X and y is an element of set Y Question 112 What s an injection, also known as an injective relation
Answer 112 Imagine two sets X and Y Suppose that a relation assigns each element of X to exactly one element of Y Also suppose that, according to the same relation, an element of Y never has more than one mate in X (Some elements of Y might have no mates in X) In a situation like this, the relation is an injection Question 113 What s a surjection, also called an onto relation
Review Questions and Answers Answer 113 Again, imagine two sets X and Y Suppose that according to a certain relation, every element of Y has at least one (and maybe more than one) mate in X, so that no element of Y is left out A relation of this type is a surjection from X onto Y Question 114 What s a bijection, also called a onetoone correspondence
Answer 114 A bijection is a relation that s both an injection and a surjection Given two sets X and Y, a bijection assigns every element of X to exactly one element of Y, and vice versa This is why a bijection is sometimes called a onetoone correspondence Question 115 What s a twospace function Is every twospace function a relation Is every twospace relation a function Answer 115 A twospace function is a relation between two sets that never maps any element of the source set to more than one element of the destination set All twospace functions are relations However, not all twospace relations are functions Question 116 What s the verticalline test for the graph of a twospace function
Answer 116 The verticalline test is a quick way to determine, based on the graph of a twospace relation, whether or not the relation is a function Imagine an infinitely long, movable line that s always parallel to the dependentvariable axis (usually the vertical axis) Suppose that we re free to move the line to the left or right, so it intersects the independentvariable axis (usually the horizontal axis) wherever we want The graph is a function of the independent variable if and only if the movable vertical line never intersects the graph at more than one point Question 117 Based on the verticalline test, which of the curves in Fig 201 are functions of x within the span of values for which 6 < x < 6 Answer 117 Only f is a function of x If we construct a movable vertical line (always parallel to the y axis), it never intersects the curve for f at more than one point over the span of values for which 6 < x < 6 However, the movable vertical line intersects the curve for g at more than one point for some values of x where 6 < x < 6 The same is true of the curve for h Question 118 Suppose we re working in the polar coordinate plane, and we encounter the graph of a relation where the independent variable is represented by q (the direction angle) and the dependent

