a.
Explain the given set is finite.
a.
Answer to Problem 3E
The set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So cardinality of set
Hence, set
b.
Explain the given set is finite.
b.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
c.
Explain the given set is finite.
c.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
d.
Explain the given set is finite.
d.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
e.
Explain the given set is finite.
e.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
f.
Explain the given set is finite.
f.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
g.
Explain the given set is finite.
g.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of the set
Hence, set
h.
Explain the given set is finite.
h.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
Hence, set
i.
Explain the given set is finite.
i.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So the cardinality of set
It implies that cardinality of set
Hence, set
j.
Explain the given set is finite.
j.
Answer to Problem 3E
Set
Explanation of Solution
Given information:
Calculation:
Consider, definition and result for finite set,
Definition: the set
Result:
If a set
The given set is as,
So
The maximum cardinality of set
Hence, the cardinality for given set
Hence, set
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Chapter 5 Solutions
A Transition to Advanced Mathematics
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward
- Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]arrow_forwardA Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.arrow_forwarda. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.arrow_forward
- Label each of the following statements as either true or false. 1. Mapping composition is a commutative operation.arrow_forwardLet R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forward
- Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardLet G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning