By Peter J. Cameron

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**Additional resources for Introduction to Algebra, 2nd edition**

**Example text**

We want to show that E(b) = E(c); by the test for equality of sets, we must show that every element of E(b) is in E(c) and vice versa. So take x ∈ E(b). Then (b, x) ∈ R. Since (c, b) ∈ R, (Eq3) gives (c, x) ∈ R, so x ∈ E(c). The reverse implication is similar. (b) Now let {A1 , A2 , . } be a partition of A. Deﬁne a relation R on A by the rule R = {(a, b) : a, b ∈ Ai for some i}. This relation R is • • • reﬂexive: for, given a ∈ A, some set Ai contains a, and so a, a ∈ Ai , whence (a, a) ∈ R; symmetric, trivially; transitive: for suppose that (a, b), (b, c) ∈ R.

In modern terminology his solution is x= 52 + 39 − 5 = 3. What is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root 30 Introduction of this square, which itself, of course is 9.

Is the converse true? Modular Arithmetic In this section we deﬁne the arithmetic of ‘integers mod m’ for any positive integer m. First, we look at Euclid’s Algorithm. 21 Euclid’s Algorithm. , of two positive integers m and n is the largest positive integer which divides both. We write it as gcd(m, n). Thus, gcd(12, 18) = 6. We can extend the notion of greatest common divisor to the case where one of the integers is equal to 0. Since any positive integer divides 0, we see that gcd(m, 0) = m if m = 0.