Let
Exercise 15c.
c. For this same
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Elements Of Modern Algebra
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forward27. 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 .arrow_forwardLet be as described in the proof of Theorem. Give a specific example of a positive element of .arrow_forward
- 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)={ 2x1ifxiseven2xifxisoddarrow_forwardProve statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.arrow_forward2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .arrow_forward
- 3. For each of the following mappings, write out and for the given and, where.arrow_forwardFor 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.arrow_forwardComplete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .arrow_forward
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