Introduction to discrete math solution
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. (Source: wikipedia)
I like to share this Power Set of the empty Set with you all through my article.
Discrete math solution example problem
Example 1:
Prove that (Z, +) is an infinite abelian group.
Solution:
(i) Closure axiom:
We know that sum of two integers is again an integer.
(ii) Associative axiom:
Addition is always associative in Z i.e., ?a, b, c ? Z, (a + b) + c = a + (b + c)
(iii) Identity axiom:
The identity element O ? Z and it satisfies O + a = a + O = a, ? a ? Z Identity axiom is true.
(iv) Inverse axiom:
For every a ? Z, ? an element - a ? Z such that - a + a = a + (- a) = 0
Therefore Inverse axiom is true.
Therefore (Z, +) is a group.
(v) ? a, b ? Z, a + b = b + a
Therefore Addition is commutative. ? (Z, +) is an abelian group.
(vi) Since Z is an infinite set (Z, +) is infinite abelian group.
Example 2: Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + b - ab for all a, b ? G. Show that (G, *) is an infinite abelian group.
Solution: Let G = Q - {1}
Let a, b ? G. Then a and b are rational numbers and a ? 1, b ? 1.
(i) Closure axiom: Clearly a * b = a + b - ab is a rational number. But to prove a * b ? G, we have to prove that a * b ? 1.
On the contrary, assume that a * b = 1 then
a + b - ab = 1
? b - ab = 1 - a
? b(1 - a) = 1 - a
? b = 1 (‡ a ? 1, 1- a ? 0)
This is impossible, because b ? 1. ? Our assumption is wrong.
Therefore a * b ? 1 and hence a * b ? G.
Therefore Closure axiom is true.
(ii) Associative axiom:
a * (b * c) = a * (b + c - bc)
= a + (b + c - bc) - a (b + c - bc)
= a + b + c - bc - ab - ac + abc
(a * b) * c = (a + b - ab) * c
= (a + b - ab) + c - (a + b - ab) c
= a + b + c - ab - ac - bc + abc
Therefore a * (b * c) = (a * b) * c ? a, b, c ? G
Therefore Associative axiom is true.
(iii) Identity axiom: Let e be the identity element.
By definition of e, a * e = a
By definition of *, a * e = a + e - ae
? a + e - ae = a
? e(1 - a) = 0
? e = 0 since a ? 1
e = 0 ? G
Therefore Identity axiom is satisfied.
(iv) Commutative axiom:
For any a, b ? G, a * b = a + b - ab
= b + a - ba
= b * a
Therefore * is commutative in G and hence (G, *) is an abelian group. Since G is infinite, (G, *) is an infinite abelian group. Please express your views of this topic math tutor free online by commenting on blog.
Discrete math solution practice problem
Problem 1:
Show that the set G = {2n / n ? Z} is an abelian group under the multiplications.
Problem 2:
Show that the set of all positive even integers forms a semi-group under the usual addition and multiplication. Is it a monoid under each of the above operations?
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. (Source: wikipedia)
I like to share this Power Set of the empty Set with you all through my article.
Discrete math solution example problem
Example 1:
Prove that (Z, +) is an infinite abelian group.
Solution:
(i) Closure axiom:
We know that sum of two integers is again an integer.
(ii) Associative axiom:
Addition is always associative in Z i.e., ?a, b, c ? Z, (a + b) + c = a + (b + c)
(iii) Identity axiom:
The identity element O ? Z and it satisfies O + a = a + O = a, ? a ? Z Identity axiom is true.
(iv) Inverse axiom:
For every a ? Z, ? an element - a ? Z such that - a + a = a + (- a) = 0
Therefore Inverse axiom is true.
Therefore (Z, +) is a group.
(v) ? a, b ? Z, a + b = b + a
Therefore Addition is commutative. ? (Z, +) is an abelian group.
(vi) Since Z is an infinite set (Z, +) is infinite abelian group.
Example 2: Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + b - ab for all a, b ? G. Show that (G, *) is an infinite abelian group.
Solution: Let G = Q - {1}
Let a, b ? G. Then a and b are rational numbers and a ? 1, b ? 1.
(i) Closure axiom: Clearly a * b = a + b - ab is a rational number. But to prove a * b ? G, we have to prove that a * b ? 1.
On the contrary, assume that a * b = 1 then
a + b - ab = 1
? b - ab = 1 - a
? b(1 - a) = 1 - a
? b = 1 (‡ a ? 1, 1- a ? 0)
This is impossible, because b ? 1. ? Our assumption is wrong.
Therefore a * b ? 1 and hence a * b ? G.
Therefore Closure axiom is true.
(ii) Associative axiom:
a * (b * c) = a * (b + c - bc)
= a + (b + c - bc) - a (b + c - bc)
= a + b + c - bc - ab - ac + abc
(a * b) * c = (a + b - ab) * c
= (a + b - ab) + c - (a + b - ab) c
= a + b + c - ab - ac - bc + abc
Therefore a * (b * c) = (a * b) * c ? a, b, c ? G
Therefore Associative axiom is true.
(iii) Identity axiom: Let e be the identity element.
By definition of e, a * e = a
By definition of *, a * e = a + e - ae
? a + e - ae = a
? e(1 - a) = 0
? e = 0 since a ? 1
e = 0 ? G
Therefore Identity axiom is satisfied.
(iv) Commutative axiom:
For any a, b ? G, a * b = a + b - ab
= b + a - ba
= b * a
Therefore * is commutative in G and hence (G, *) is an abelian group. Since G is infinite, (G, *) is an infinite abelian group. Please express your views of this topic math tutor free online by commenting on blog.
Discrete math solution practice problem
Problem 1:
Show that the set G = {2n / n ? Z} is an abelian group under the multiplications.
Problem 2:
Show that the set of all positive even integers forms a semi-group under the usual addition and multiplication. Is it a monoid under each of the above operations?
Posts
No comments:
Post a Comment