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- 30. Prove statement of Theorem : for all integers .21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.31. Prove statement of Theorem : for all integers and .
- Let be as described in the proof of Theorem. Give a specific example of a positive element of .A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers. a. if and only if . b. if and only if . c. if and only if for some in . d. if and only if . e. if and only if . f. if and only if . g. if and only if . h. if and only if . i. if and only if . j. if and only if . k. if and only if .Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .
- Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.Find all monic irreducible polynomials of degree 2 over Z3.Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.