By Minorsky N.

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The fact that it has no elements does not mean that it has any less ‘existence’ than other sets, any more than zero has less existence than the positive numbers. 1 (1) with solution Show that ¿ Â A for every set A. Solution We need to show that for all x, if x 2 ¿, then x 2 A. In other words, there is no x with x 2 ¿ but x 62 A. But by the definition of ¿, there is no x with x 2 ¿, so we are done. 4 Boolean Operations on Sets 11 Alice Box: If : : : then : : : Alice: That’s a short proof, but a strange one.

E. we may have A B ¤ B A), the operation of counting the elements of the Cartesian product is commutative: always #(A B) D #(B A). 2 (2) (with partial solution) (a) Show that when A Â A0 and B Â B0 , then A B Â A0 B0 . (b) Show that when both A ¤ ¿ and B ¤ ¿, then A B D B A iff A D B. 30 2 Comparing Things: Relations Solution to (b) Suppose A ¤ ¿ and B ¤ ¿. We need to show that A B D B A iff A D B. We do this in two parts. First, we show that if A D B, then A B D B A. Suppose A D B. By the supposition, A B D A2 and also B A D A2 , so that A B D B A as desired.

Given a relation R, we define R 1 to be the set of all ordered pairs (b,a) such that (a,b) 2 R. Warning: Do not confuse this with complementation! There is nothing negative about conversion despite the standard notation R 1 : we are simply reversing the direction of the relation. The complement of loves is doesn’t love, but its converse is is loved by. 1 (1) (with solutions) (a) Let A be the set of natural numbers. What are the converses of the following relations over A: (i) less than, (ii) less than or equal to, (iii) equal to.