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July 4th, 2018, 11:28 PM   #1
Joined: Jul 2018
From: india

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Post Immediate Help Required in discrete mathematics

Hi Friends,

I am new to this forum and my son is awfully struck in some discrete mathematics question. Your help is highly solicited. The questions are posted below:

1. Prove that if H, K are subgroups of a group G and H Ų K = G. Then either H=G or K=G

2. Let G be a group and a, b Є G. Then the equation x*a=b has a unique solution given by x= b* a

3. Linear sum W1+ W2 of two subspaces W1and W2 of a vector space V(F) is A subspace of V(F)

4. show that the function T: R2 → R2 such that T(0,1)=(3,4),T(3,1)=(2,2) And T(3,2)=(5,7) is not a L.T.

5. Let T:v →w be a linear transformation. Then T is onto iff p(T)= dim w.

6. show that the function T: R2 → R2 defined by T(x1,x2)=(x1-x2,x1+x2),for (x1,x2)Є R2 is bijective

Please provide detailed steps as I do not know these mathematics, but when I give this to my son, he can understand. Thanks in advance friends.
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July 5th, 2018, 02:15 AM   #2
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2. For a, b ∈ G, where (G,*) is a group, let c ∈ G be the unique inverse of a,
then x*a = b implies (x*a)*c = b*c, which simplifies to x = b*c.

Note: the usual axioms of a group (G,*) specify that the group operation is associative and that each element of G has an inverse. It's an elementary theorem that no element has more than one inverse.

4. If T is a L.T., T(0,1) + T(3,1) = T(0+3,1+1) = T(3,2) = (5,7),
but T(0,1) + T(3,1) = (3,4) + (2,2) = (5,6), not (5,7).

6. T is bijective as it has an inverse Q(x,y) = ((x+y)/2,(y-x)/2).
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