Changes

::::::: I can't vouch for this on a national level, but when I took the Putnam at UConn last year, about 90% of the participants were male. If that's true at a national level (and I don't know if it is) then it stands to reason. [[User:Gregkochuconn|Gregkochuconn]] 21:28, 26 February 2012 (EST)

::::::: I can't vouch for this on a national level, but when I took the Putnam at UConn last year, about 90% of the participants were male. If that's true at a national level (and I don't know if it is) then it stands to reason. [[User:Gregkochuconn|Gregkochuconn]] 21:28, 26 February 2012 (EST)

::::::: It's gotten so bad that now they make women put red stickers on the exams so they can graded differently!  Women often have separate prizes too.--Bogart12

::::::: It's gotten so bad that now they make women put red stickers on the exams so they can graded differently!  Women often have separate prizes too.--Bogart12

::::::::I don't know why anyone (i.e., liberals) would expect or pretend that God created men and women to have absolutely identical aptitudes in everything.  Men and women are physically very different.--[[User:Aschlafly|Andy Schlafly]] 01:06, 27 February 2012 (EST)

::::::::I don't know why anyone (i.e., liberals) would expect or pretend that God created men and women to have absolutely identical aptitudes in everything.  Men and women are physically very different.--[[User:Aschlafly|Andy Schlafly]] 01:06, 27 February 2012 (EST)

::::But hammers are not courses of study or an ideology.  Math is.  And all courses of study and ideologies are susceptible to liberal bias.  Engineering, with its grounding in what works, is probably the most immune to [[liberal bias]].  Women's studies and English literature are probably the most susceptible to liberal bias.--[[User:Aschlafly|Andy Schlafly]] 22:37, 26 February 2012 (EST)

::::But hammers are not courses of study or an ideology.  Math is.  And all courses of study and ideologies are susceptible to liberal bias.  Engineering, with its grounding in what works, is probably the most immune to [[liberal bias]].  Women's studies and English literature are probably the most susceptible to liberal bias.--[[User:Aschlafly|Andy Schlafly]] 22:37, 26 February 2012 (EST)

== atheist and liberal idiocy extends to math: 2+2=5? (Lawrence Krauss vs William Lane Craig) ==

== atheist and liberal idiocy extends to math: 2+2=5? (Lawrence Krauss vs William Lane Craig) ==

{| class="wikitable"

{| class="wikitable"

|-

|-

! \cdot

! <math>\cdot</math>

! <math>a</math>

! <math>a</math>

! <math>b</math>

! <math>b</math>

| <math>c</math>

| <math>c</math>

|}

|}

:*The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition

:*The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition.

:*The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.

:*The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1, and <math>\cdot</math> is standard multiplication of complex numbers.

:By definition, none of these groups are equal, since groups are technically ordered pairs and none of the underlying sets are equal.  However, each pair of these groups is isomorphic, which means that there exists a bijective homomorphism.  Isomorphic groups have the same group structure.  For example, each of these groups is abelian, which means that the group operation is commutative.  Also, each group has four elements.  Further, each group is cyclic, meaning that it is generated by a single element; the Klein four-group is not cyclic.  Therefore, when we say that there are only two groups of order 4 ''up to isomorphism'', we use isomorphisms and homomorphisms because we really don't care about groups that have the same structure but just have the elements relabeled.

:By definition, none of these groups are equal, since groups are technically ordered pairs and none of the underlying sets are equal.  However, each pair of these groups is isomorphic, which means that there exists a bijective homomorphism.  Isomorphic groups have the same group structure.  For example, each of these groups is abelian, which means that the group operation is commutative.  Also, each group has four elements.  Further, each group is cyclic, meaning that it is generated by a single element; the Klein four-group is not cyclic.  Therefore, when we say that there are only two groups of order 4 ''up to isomorphism'', we use isomorphisms and homomorphisms because we really don't care about groups that have the same structure but just have the elements relabeled.

:Homomorphisms exist between any pair of groups: the most obvious example is the homomorphism mapping every element of the domain group to the identity element of the target group.  Homomorphisms are very interesting for many reasons; the kernel of a homomorphism (the preimage of the identity element) is always a normal subgroup, and free groups <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)

:Homomorphisms exist between any pair of groups: the most obvious example is the homomorphism mapping every element of the domain group to the identity element of the target group.  Homomorphisms are very interesting for many reasons; the kernel of a homomorphism (the preimage of the identity element) is always a normal subgroup, and free groups <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)