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Extra resources for Schaum's Outline of Discrete Mathematics, Revised Third Edition (Schaum's Outline Series)
Example text
C) List all equivalence classes with more than one element. 32. Let S = {1, 2, 3, . . , 14, 15}. Let R be the equivalence relation on S defined by x ≡ y (mod 5), that is, x − y is divisible by 5. e. the quotient set S/R. 33. Let S = {1, 2, 3, . . , 9}, and let ∼ be the relation on A × A defined by (a, b) ∼ (c, d) whenever a + d = b + c. (a) Prove that ∼ is an equivalence relation. (b) Find [(2, 5)], that is, the equivalence class of (2, 5). 20. 21. (a) x = 3, y = −2; (b) x = 2, y = 3. 23. (a) MR = [0, 0, 1, 1; 0, 0, 0, 0; 0, 1, 1, 1; 0, 0, 0, 0]; (b) Domain = {1, 3}, range = {2, 3, 4}; (c) R −1 = {(3, 1), (4, 1), (2, 3), (3, 3), (4, 3)}; (d) See Fig.
Inverse Relation Let R be any relation from a set A to a set B. The inverse of R, denoted by R −1 , is the relation from B to A which consists of those ordered pairs which, when reversed, belong to R; that is, R −1 = {(b, a) | (a, b) ∈ R} For example, let A = {1, 2, 3} and B = {x, y, z}. Then the inverse of R = {(1, y), (1, z), (3, y)} is R −1 = {(y, 1), (z, 1), (y, 3)} Clearly, if R is any relation, then (R −1 )−1 = R. Also, the domain and range of R −1 are equal, respectively, to the range and domain of R.
1-5(a) shows sets A, B, C. Shade the following sets: (a) A\(B ∪ C); (b) AC ∩ (B ∪ C); (c) AC ∩ (C\B). 35 Use the Venn diagram in Fig. 1-5(b) to write each set as the (disjoint) union of fundamental products: (a) A ∩ (B ∪ C); (b) AC ∩ (B ∪ C); (c) A ∪ (B\C). 36 Consider the following assumptions: S1 : All dictionaries are useful. S2 : Mary owns only romance novels. S3 : No romance novel is useful. Use a Venn diagram to determine the validity of each of the following conclusions: (a) Romance novels are not dictionaries.