What s a Two-Space Relation 213

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For every value of x, the relation assigns one and only one value of y The converse is also true; for every value of y, there is one and only one corresponding value of x

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Example 2 Next, let s consider a real-number relation that squares each element in the domain to produce values in the range We can write this relation as the following equation:

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y = x2 In the set of real numbers, this relation is defined for all possible values of x, but we never get any negative values of y The range is the set of all y such that y 0 When we plug specific numbers into this equation, we get results such as the following: If x = 4, then (x,y) = ( 4,16) If x = 1, then (x,y) = ( 1,1) If x = 1/2, then (x,y) = ( 1/2,1/4) If x = 0, then (x,y) = (0,0) If x = 1/2, then (x,y) = (1/2,1/4) If x = 1, then (x,y) = (1,1) If x = 4, then (x,y) = (4,16)

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For every value of x, the relation assigns a unique value of y, but for every assigned value of y except y = 0 in the range, the domain contains two values of x

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Example 3 Now let s look at a real-number relation that takes the positive or negative square root of elements in the domain to get elements in the range We can write it as the equation

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y = (x1/2) When we plug in some numbers here, we get results like the following: If x = 1/9, then (x,y) = (1/9,1/3) or (1/9, 1/3) If x = 1/4, then (x,y) = (1/4,1/2) or (1/4, 1/2) If x = 1, then (x,y) = (1,1) or (1, 1) If x = 4, then (x,y) = (4,2) or (4, 2) If x = 9, then (x,y) = (9,3) or (9, 3) If x = 0, then (x,y) = (0,0)

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In the set of real numbers, the domain of this relation is confined to nonnegative values of x That is, the domain is the set of all x such that x 0 For every positive value of x in the domain, there are two values of y in the range If x = 0, then y = 0 The range encompasses all possible real-number values of y For any value of y in the range, there exists one and only one corresponding value of x in the domain

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Relations in Two-Space

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Example 4 Finally, let s examine a real-number relation that takes the nonnegative square root of values in the domain to get values in the range We can denote it as

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y = x1/2 The domain is the set of all real numbers x such that x 0, and the range is the set of all real numbers y such that y 0 Following are a few examples of what happens when we input values of x into this equation: If x = 1/9, then (x,y) = (1/9,1/3) If x = 1/4, then (x,y) = (1/4,1/2) If x = 1, then (x,y) = (1,1) If x = 4, then (x,y) = (4,2) If x = 9, then (x,y) = (9,3) If x = 0, then (x,y) = (0,0)

For every x in the domain, there is one and only one y in the range The converse is also true For every y in the range, there is one and only one x in the domain

Are you confused

Sometimes a relation fails to take all of the elements of the source or destination sets into account Figure 11-1 illustrates a generic example of a situation of this sort using a graphical scheme called a Venn diagram: The entire source set is called the maximal domain The entire destination set is called the co-domain The domain of a relation is a subset of its maximal domain The range of a relation is a subset of its co-domain

Here s a challenge!

Classify each of the relations in Examples 1 through 4 as an injection, a surjection, a bijection, or none of them from the set of real numbers to itself