By Seymour Lipschutz

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**Additional resources for Schaum's Outline of Discrete Mathematics, Revised Third Edition (Schaum's Outline Series)**

**Sample text**

By the principle of induction, P (n) is true for all n. 26 Which of the following sets are equal? A = {x | x 2 − 4x + 3 = 0}, B = {x | x 2 − 3x + 2 = 0}, C = {x | x ∈ N, x < 3}, D = {x | x ∈ N, x is odd, x < 5}, E = {1, 2}, F = {1, 2, 1}, G = {3, 1}, H = {1, 1, 3}. 27 List the elements of the following sets if the universal set is U = {a, b, c, …, y, z}. Furthermore, identify which of the sets, if any, are equal. A = {x | x is a vowel}, B = {x | x is a letter in the word “little”}, C = {x | x precedes f in the alphabet}, D = {x | x is a letter in the word “title”}.

B) Draw the arrow diagram of R. (c) Find the inverse relation R −1 of R. (d) Determine the domain and range of R. (a) See Fig. 2-6(a) Observe that the rows of the matrix are labeled by the elements of A and the columns by the elements of B. Also observe that the entry in the matrix corresponding to a ∈ A and b ∈ B is 1 if a is related to b and 0 otherwise. (b) See Fig. , iff (a, b) ∈ R. CHAP. 2] RELATIONS 35 Fig. 2-6 (c) Reverse the ordered pairs of R to obtain R −1 : R −1 = {(y, 1), (z, 1), (y, 3), (x, 4), (z, 4)} Observe that by reversing the arrows in Fig.

Two ordered pairs are equal if and only if the corresponding components are equal. Hence we obtain the equations 2x = 6 and x+y =2 from which we derive the answers x = 3 and y = −1. 3. Find the number of relations from A = {a, b, c} to B = {1, 2}. There are 3(2) = 6 elements in A × B, and hence there are m = 26 = 64 subsets of A × B. Thus there are m = 64 relations from A to B. 4. Given A = {1, 2, 3, 4} and B = {x, y, z}. Let R be the following relation from A to B: R = {(1, y), (1, z), (3, y), (4, x), (4, z)} (a) Determine the matrix of the relation.