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**Additional resources for Schaums Outline of Discrete Mathematics, Revised Third Edition**

**Example text**

38 Use the laws in Table 1-1 to prove each set identity: (a) (A ∩ B) ∪ (A ∩ B C ) = A (b) A ∪ B = (A ∩ B C ) ∪ (AC ∩ B) ∪ (A ∩ B) 20 SET THEORY [CHAP. 39 Determine which of the following sets are ﬁnite: (a) Lines parallel to the x axis. (c) Integers which are multiples of 5. (b) Letters in the English alphabet. (d) Animals living on the earth. 10: Suppose A, B, C are ﬁnite sets. 41 A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of three popular options, air-conditioning (A), radio (R), and power windows (W ), were already installed.

16) that P (R) = ∩(S | S is a P -relation and R ⊆ S) Thus one can obtain P (R) from the “top-down,” that is, as the intersection of relations. However, one usually wants to ﬁnd P (R) from the “bottom-up,” that is, by adjoining elements to R to obtain P (R). This we do below. Reﬂexive and Symmetric Closures A The next theorem tells us how to obtain easily the reﬂexive and symmetric closures of a relation. Here = {(a, a) | a ∈ A} is the diagonal or equality relation on A. 3: Let R be a relation on a set A.

2-7 36 RELATIONS [CHAP. 2 (b) The matrices of MR , MS , and MR◦S follow: a 0 1 MR = 2 1 0 3 b 1 0 0 c 0 1 0 x 0 a MS = b 1 0 c y 1 0 1 z 0 0 1 x 1 1 MR◦S = 2 0 0 3 y 0 1 0 z 0 1 0 Multiplying MR and MS we obtain 1 MR MS = 0 0 0 2 0 0 1 0 Observe that MR◦S and MR MS have the same zero entries. 6. Consider the relation R = {(1, 1), (2, 2), (2, 3), (3, 2), (4, 2), (4, 4)} on A = {1, 2, 3, 4}. (a) Draw its directed graph. (b) Find R 2 = R◦R. (a) For each (a, b) ∈ R, draw an arrow from a to b as in Fig.