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We can figure all of these out by looking at Fig 9-2 The answers, bullet-by-bullet, are as follows Set Q is entirely contained in set R Therefore, all rationals are real Set Z is entirely contained in set R Therefore, all integers are real Set Z is completely separate from set S Therefore, no integers are irrational Set S has elements in common with set R Therefore, some irrationals are real

Lastly, eqn (567) can be manipulated as shown below: P+ = [I KH] P + P H K P H K = [I KH] P + K R + HP H K P H K = P KHP + KRK + KHP H K P H K = [I KH] P [I KH] + KRK (569) where eqn (566) was used in the transition from the rst to second line Eqn (569) is referred to as the Joseph form of the covariance propagation equations [34]

S2,S4 SPST toggle switches S3 3-position rotary switch, break before make 100-micro-ampere moving-coil panelmounted meter

Let s explore infinity for a few minutes Then we ll prove that the square root of 2 is irrational Get into the mood for some serious abstract thinking!

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This section presents a few Kalman lter examples The following example is useful in various applications including the estimation of the GPS ionospheric error using data from a two frequency receiver as is discussed in Section 86 and carrier smoothing of code measurements as is discussed in Section 87 Example 54 Consider the scenario where at each measurement epoch two measurements are available yk zk = = xk + nk xk + vk + B (570) (571)

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Number lists Mathematicians use the symbol 0 (called aleph-null ) to describe the number of elements in the set N of natural numbers This is the same as the number of elements in the set Z of integers, as we saw in the solution to the final practice exercise in Chap 3 We can create implied lists of both sets, and be confident that if we go far enough out, we ll always hit any natural number or integer we care to choose Cardinality of a set The number of elements in a set is called the cardinality of the set The cardinality of N is 0, and the cardinality of Z is also 0 Even the elements of Q, the set of rationals, can be defined in terms of an implied list Figure 9-4 is an example So, as counterintuitive as it may seem, the cardinality of Q is the same as the cardinality of N or Z, that is, 0 If we can make an implied list of the elements in an infinite set, that set is said to be denumerably infinite (or simply denumerable), and by definition it has cardinality 0 The irrationals and reals can t be listed The elements of S or R cannot be denoted in any type of list We can t even make an implied list of all the nonnegative irrational numbers smaller than 1 in decimal-expansion form Table 9-1, and the following explanation, should give you some idea of why this is so Suppose we try to list the irrational numbers between 0 and 1 (including 0, but not 1) by writing down the numerals in their expanded-decimal forms To the left of the decimal point, every numeral will have a single 0 and nothing else The first numeral will have an endless string of digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to the right of the decimal point Let s call those digits a11, a12, a13, and so on The second number will have an endless string of digits that we can call a21, a22, a23, and so on, different from the first string We can keep on listing irrational numbers like this forever The nth number in our list (corresponding to the nth row in Table 9-1) will be of the form.

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