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Let B be the set of all binary strings of length 2; i.e. B={ (0,0), (0,1),...

Let B be the set of all binary strings of length 2; i.e. B={ (0,0), (0,1), (1,0), (1,1)}. Define the addition and multiplication as coordinate-wise addition and multiplication modulo 2. It turns out that B becomes a Boolean algebra under those two operations. Show that B under addition is a group but B under multiplication is not a group.

Coordinate-wise addition and multiplication modulo 2 means (a,b)+(c,d)=(a+c, b+d), (a,b)(c,d)=(ac, bd), in addition to the fact that 1+1=0.

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