Abstract algebra: Take a group G, and a group H, with a group operation *. G is isomorphic to H if there exist a map f: G->H such that:
1: f is injective
2: f is surjective
3: f(a*b)=f(a)*f(b) for all a,b in G
If f satisfies these three properties, f is called an isomorphism.
1: f is injective
2: f is surjective
3: f(a*b)=f(a)*f(b) for all a,b in G
If f satisfies these three properties, f is called an isomorphism.
The map f: Z -> E given by f(a)=2a where Z is the integers and E is the even integers is an isomorphism.
Proof:
Showing injectivity
f(b)=f(a) => 2a=2b (from the given function) <=> a=b
Showing surjectivity
Suppose n is in E. n is an even integer hence n=2k for some integer k.
f(k)=2k=n, hence f is surjective.
Homomorphism:
f(a+b)=2(a+b)=2a+2b=f(a)+f(b)
Hence f is an isomorphism. Q. E. D.
Proof:
Showing injectivity
f(b)=f(a) => 2a=2b (from the given function) <=> a=b
Showing surjectivity
Suppose n is in E. n is an even integer hence n=2k for some integer k.
f(k)=2k=n, hence f is surjective.
Homomorphism:
f(a+b)=2(a+b)=2a+2b=f(a)+f(b)
Hence f is an isomorphism. Q. E. D.
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by Randyt13 June 22, 2013
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like isotopes, which are two different atoms (with different atomic mass number) having the same atomic number.
like Mayank & Paremal having their birthday on the same day, call each other:
like isotopes, which are two different atoms (with different atomic mass number) having the same atomic number.
like Mayank & Paremal having their birthday on the same day, call each other:
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Get the Isko mug.Each element has many versions that contain a different number of neutrons in the nucleas. Each different version of the element is called an isotope.
Carbon-13 is and isotope of Carbon
by PrivateDonut February 10, 2005
Get the Isotope mug.1. A big badass nigga who dont give a damn abt nuthin and will smoke any mf who gets in his way. #Naija ninja lexicon hard guy, hardges, hard g
2. Used to describe an accomplished man, usually selfmade, who has become a pillar of his society. Southern Nigerian in origin made man
3. A large hardwood tree from west coast of tropical Africa. It is a very durable wood; iroko does not require regular treatment with oil or varnish when used outdoors. Although it is almost as durable as teak, it does not have the same stability. A tree native to SW Nigeria. African teak Milicia excelsa Chlorophora excelsa Chlorophora regia
4. Furniture and other woodwork that comes from the Iroko tree
5. The Iroko is sacred to the Yoruba people. The tree is feared and hence is shunned or revered with offerings. Yoruba people believe that anybody who sees the Iroko-man face to face who is said to inhabit becomes insane and speedily dies. According to the Yoruba, any man who cuts down any Iroko tree results in devastating misfortune on the woodcutter and all of his family. They also claim that the spirit of the Iroko can be heard in houses which use Iroko wood and that it is the spirit of the Iroko, trapped in the wood idol tree spirit
2. Used to describe an accomplished man, usually selfmade, who has become a pillar of his society. Southern Nigerian in origin made man
3. A large hardwood tree from west coast of tropical Africa. It is a very durable wood; iroko does not require regular treatment with oil or varnish when used outdoors. Although it is almost as durable as teak, it does not have the same stability. A tree native to SW Nigeria. African teak Milicia excelsa Chlorophora excelsa Chlorophora regia
4. Furniture and other woodwork that comes from the Iroko tree
5. The Iroko is sacred to the Yoruba people. The tree is feared and hence is shunned or revered with offerings. Yoruba people believe that anybody who sees the Iroko-man face to face who is said to inhabit becomes insane and speedily dies. According to the Yoruba, any man who cuts down any Iroko tree results in devastating misfortune on the woodcutter and all of his family. They also claim that the spirit of the Iroko can be heard in houses which use Iroko wood and that it is the spirit of the Iroko, trapped in the wood idol tree spirit
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2. Chief Awolowo was a colossus that bestrode his times; he was a real Iroko. made man
There are several untapped wild (natural-growing) iroko stands in Yoruba country in Nigeria
4. My big iroko desk
There is a general office furniture upgrade to iroko.
The parlour was done in period iroko (precolonial regal decor)
After the Irokoman granted Oluronbi phenomenal sales in the market, he requested for her only daughter which she had earlier promised, and he wouldn't accept anything else
(From the popular Yoruba folktale, Oluronbi) idol tree spirit
2. Chief Awolowo was a colossus that bestrode his times; he was a real Iroko. made man
There are several untapped wild (natural-growing) iroko stands in Yoruba country in Nigeria
4. My big iroko desk
There is a general office furniture upgrade to iroko.
The parlour was done in period iroko (precolonial regal decor)
After the Irokoman granted Oluronbi phenomenal sales in the market, he requested for her only daughter which she had earlier promised, and he wouldn't accept anything else
(From the popular Yoruba folktale, Oluronbi) idol tree spirit
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