Elementary Set Theory: Pt. 1 by Kam-Tim Leung, Doris Lai-Chue Chen

By Kam-Tim Leung, Doris Lai-Chue Chen

This e-book presents scholars of arithmetic with the minimal volume of information in good judgment and set conception wanted for a ecocnomic continuation in their experiences. there's a bankruptcy on assertion calculus, via 8 chapters on set thought.

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3. 4. empty set. The unique set which contains no element is called the The empty set is also called the void set or the null sety and is denoted throughout this book by the symbol 0 . Any set which is not the empty set is called a non-empty set. Clearly the only subset of 0 is 0 itself. Moreover, 0 is also characterized by the property that 0 is a subset of any set. This means that (i) for any set A, 0 c At and (ii) if a set B is such that B c A for every set Ay then B = 0 . Since xe0 is always false, the conditional xe0 -* xeA for every set A is always true.

A\B 18. Prove the following statements for any two subsets A and B of a set E: (a) B\A<^E\A. (b) B\(E\A) = B (\A. (c) If A n B = 0, then B n ( £ V 4 ) = B and ,4 U ( £ \ B ) = £ \ B . (d) (£ V 4 ) \ ( £ \ B ) = B\A. (e) If ^ U B = £ and A fl B = 0 then B = £ \ , 4 . 19. Is the statement A(\B = 0 iff (E\A) n (E\B) = 0 true for any two subsets A and B of a set E? If not, illustrate by a counter example. 20. Determine whether the following statements are true for any three sets Ay By C: (a) A\(B\C) = (A\B)\C (b) A n (B\C) = (Ao B)\(A n C).

Cz D. -related to br) for (ayb)eGy where R = (A,B,G). D. Inverses and compositions Intersection and union of two relations Rx — (AyB,Gx) and R2 = (AyByG2) having the same set of departure and the same set of destination are defined as follows: RX()R2 = (AyByGx n G2) # ! U R2 = (AyByG^ u G2) But we shall not discuss these further, as they are seldom used in mathematics. The more important operations on relations are the forming of inverses and compositionSy which we introduce now. 8. If R = (AyB,G) is a relation, then the inverse relation of the relation R is the relation R1 = (ByAyG~l) where G-i = {(bta)eB x A:(ayb)eG} Notice that the set of departure of R is the set of destination of R'1 and that the set of destination of R is the set of departure of R1.

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