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By P. B. Bhattacharya

This booklet gets you there in the event you think in it. It has examples with options and issues of suggestions. the one subject that doesn't have issues of ideas is different types. For this, i've got the Hungerford textual content, and i'm almost immediately within the strategy of discovering a greater publication for this. differently it's the ideal booklet for self-study.

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Therefore, E(a) = E(b). This proves that each a in X belongs to exactly one equivalence class, namely, E(a). 4, X/E is a partition of X. Conversely, let it be a given partition of I. Define the relation E as follows. For any x,y X, x E y ifand only ifx,yare in the same block of it. Clearly, E is an equivalence relation on X, and the equivalence classes of E are the blocks of it. Hence, X/E = it. Suppose E' is also an equivalence relation on X such that X/E' = it. Then for all x,y E X, xEy x,y are in the same block of it xE' y.

For any mapping X —' X, let = o o ... • 4. (n times)1, and let 4° denote the identity mapping. Theorem (Shröder—Bernstein). 1 A —' B andg: B —'A be infective mappings. Then there exists a bijection w: A—'B. Proof 1ff is not suijective, let C= B — Set B* = c E C, n 0) and 4* = g(B) = (g(b)fb E B). Imf 27 Cardinality of a set The restriction of g to B is obviously a bijection B* A. ,u is bijective. We first prove that a A* if and only iff(a) E wherec E C, B. Ifa EA, then a =g(b)forsome b E B*. Conversely, if f(a) E B, n 0.

Proof (i) Let A = (a11). B = (b11), C = (c,1) be m X n, n X p. p X q matrices, respectively. Then A(BC) and (AB)C are both in X q matrices. Let (i,j) E m X q. kbk/) = the (i,j) entry of (A B)C. This proves A(BC) = (AB)C. The proofs for (ii) and (iii) are similar to the one for (I) and are left as exercises. 0 The Kronecker delta function is defined to be ô: N X such that jfj'j ifi#j for z,j EN It is worth noting that given an n-tuple (a1 a1ô11 = MatrIces and determinants 44 Definition. The n X n matrix in which, for each (i,j) a X a, the (i,j) is called the identity matrix of order n and denoted by entry is In other words, each entry in the diagonal of I,, is 1, and every other entry is zero.

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