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**Additional resources for Schaum's outline of theory and problems of combinatorics including concepts of graph theory**

**Example text**

This is an absurdity since there is no x E 0. I 1=M C A and 0 C A. True or false? ") The following proposition is an immediate consequence of the definitions of "subset" and "equal sets," and it illustrates the way we prove two sets are equal in practice. A *S*S 0 If A and B are sets, then A = B if and only if A C B and B C A. There are two assertions being made here. (-a) If A = B, then A is a subset of B and B is a subset of A. a-) If A is a subset of B and B is a subset of A, then A = B. 4 is to say that, for two sets to be equal, it is necessary and sufficient that each be a subset of the other.

Also, when applying Property 7, we use -p instead of p. 28 Chapter 1 Yes, There Are Proofs! The symbol [BB] means that an answer can be found in the Back of the Book. 1. Verify each of the 13 properties of logical equivalence which appear in this section [BB; 1,3,5,7,9, 11,13]. 2. Simplify each of the following statements. (a) [BB] (p A q) V (-((-p) V q)) (b) (p V r) - [(q V (-r)) - ((-p) (c) [(p - q) V (q - r)] A (r -* s) r)] 3. Using truth tables, verify the following absorption properties. (a) [BB] (p (b) (pA(pVq)) V (p Aq)) p 4.

Z = {. . , -3, -2, -1, 0, 1, 2, 3, .. is the set of integers. - On other occasions, it is convenient to describe a set with so-called set builder notation. " More generally, we see Some expression I the expression has certain properties). Thus, the set of odd natural numbers could be described as in I n is an odd integer, n > O} or as {2k- I k = 1,2,3, ... } or as {2k - 1 k c NJ. The expression "k E N" is read "k belongs to N," the symbol E denoting set membership. Thus, "m c Z" simply records the fact that m is an integer.