numbers 6, and 7. example 12 included

1.6 Exercises In Exercises I and 2, tell whether the structure has the closure (b) (prime numbers, +, *) addition property with respect to the operation. 5. Show that @ is a commutative operation for sets. 1. (a) (sets, U, n, -) union 6. Using the definitions in Example 12, (a) show that O is (b) (sets, U, n, ) complement associative. (b) Show that V is associative. 2. (a) (4 x 4 matrices, +, *, ") multiplication 7. Using the definitions in Example 12, determine if the other possible distributive property holds. (b) (3 x 5 matrices, +, *, 7) transpose 8. Give the identity element, if one exists, for each binary In Exercises 3 and 4, tell whether the structure has the closure operation in the given structure. property with respect to the operation. (a) (real numbers, +, *, V ) 3. (a) (integers, +, -, *, :) division (b) (sets, U, n, ) (b) (A*, catenation) catenation (c) (10, 1), 0, V, *) as defined in Example 12 4. (a) (n x n Boolean matrices, V, A, T) meet (d) (subsets of a finite set A, @, ) 1.6 Mathematical Structures 45 9. Give the identity element, if one exists, for each binary Determine which of the following properties hold for this operation in the structure (5 x 5 Boolean matrices, V, A, structure. O). (a) Closure (b) Commutative In Exercises 10 through 16, use the structure S = (n x n diag- (c) Associative onal matrices, +, *, "). 30. Let R be as in Exercise 29. Determine which of the fol- 10. Show that S is closed with respect to addition. lowing properties hold for this structure. 11. Show that S is closed with respect to multiplication. (a) An identity elementM My Excel Grade Sheet - Is: X D21 Dropbox Folders - 22FA1- X G Chapter 7&8 Jesus Christ X Course Hero X 0321998111.pdf X + X -> C @ File | C:/Users/loren/Downloads/pdfcoffee.com_discrete-math-6th-edition-pdf-free.pdf * . . . E 0321998111.pdf 60 / 552 100% + This is one of our first examples of a proof that does not proceed directly. We assumed that there were two identity elements and showed that they were in fact the same element. Example 10 For (n x n matrices, +, *, "), I,, is the identity for matrix multiplication and the n x n zero matrix is the identity for matrix addition. If a binary operation has an identity e, we say y is a O-inverse of x if x Oy =yox=e. THEOREM 2 If O is an associative operation and x has a -inverse y, then y is unique. Proof Assume there is another O-inverse for x, say z. Then (z O x) Dy = e by = y and z O (x O y) = z De = z. Since 0 is associative, (z O x) Dy = z 0 (x Dy) and so y = z. Example 11 (a) In the structure (3 x 3 matrices, +, *, "), each matrix A = [a;; ] has a +-inverse, or additive inverse, -A = [ -a;; ]. (b) In the structure (integers, +, *), only the integers 1 and - 1 have multiplicative inverses. Example 12 Let D, V, and o be defined for the set (0, 1} by the following tables. 0 01 0 4 1 Thus 1 0 0 = 1, 0 V 1 = 0, and 10 = 0. Determine if each of the following is true for ({0, 1), 0, V, o). (a) is commutative. (b) V is associative. 7:17 PM A 9/25/2022