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Question: for each of the pairs of sets h g...

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. For each of the pairs of sets H G below, prove whether or not H is closed under the given binary operation on the set G

(c) Let H = f [0] ; [3] ; [6] ; [9] ; [12] g , G = Z15 under the binary operation +.

2. Give an example of each or give a reason why no example exists.

(a) A solution to the equation 22 x 4 mod 4 (b

) A solution to the equation 16 x = 2 mod 4

(c) Functions f : T ! W , g : W ! S , where g f is an onto function, but g is not an onto function.

(d) A subset of Z that is closed under both addition and subtraction

. 3. Find (252 ; 180) show all of your work.

4. Let a; b be nonzero integers. Show that ( a; b ) = 1 if and only if ( a + b; ab ) = 1.

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