Suppose f is continuous on R. Prove that f is uniformly continuous on any bounded subset D of R.
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- 27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .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.[Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]
- Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
- For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.28. Let where and are nonempty. Prove that has the property that for every subset of if and only if is onto. (Compare with Exercise 15c.) Exercise 15c. c. For this same and show that.Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
- For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisoddProve that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]Let g:AB and f:BC. Prove that f is onto if fg is onto.