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Modern Syllabus Algebra by D. G. H. B. Lloyd and C. Plumpton (Auth.)

By D. G. H. B. Lloyd and C. Plumpton (Auth.)

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7. It it possible for a mod b to be equal to b mod a! If so give an example and quote the relationship which must exist between a and b. If it is not possible explain why. 8. 16 2 4 and 1 6 = 1 mod 5 use the ideas of question 5 to find the remainder when 8 " is divided by 5. 9. Find the remainder when 1718 is divided by 19. 10. Use the binomial theorem to show that {a + l) m = am + 1 mod w. Hence prove by induction that provided m is prime am — a mod m. Show by means of an example that this result is not necessarily true when m is not prime (ULIE).

The symbol used is ~ placed before the symbol of the statement it negates. Thus "Rome is not burning" would be written ~x whilst "It is false that 3 2 = 9" would be ~y. The reader may as yet be puzzled by not being able to see the connection between the foregoing and a two-element Boolean algebra. The linkup comes when we note that statements are either true or false, so every statement has a TRUTH VALUE of 1 (if it is true) or 0 (if it is false). When we combine two statements x and y by means of disjunction we have noted that the compound statement will be considered true if either x or y 55 BOOLEAN ALGEBRA (or both) is true.

3. Finite Arithmetics Ignoring the minute and second hands on a clock, the hour hand and clock face together constitute a device for adding up hours. After each hour passes the device adds one more digit. Between three o'clock and four o'clock it adds 1 on to 3 to make 4, then 1 on to 4 to make 5 and so on. However, when it comes to 11 it adds 1 to make 12 and after that 1 more to make not 13 but 1 again. If we regard 12 as the 0 of this system we have an example of a finite arithmetic, a number system with a finite number of elements.

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