3. (4 points each) Prove or disprove: (a) f : Z Z X Z defined by f...

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3. (4 points each) Prove or disprove: (a) f : Z Z X Z defined by f(n) = (n + 1, 2n) for any 72 € Z is a surjective function. 6. (8 points) Let X be a subset of N14. Prove that if IX] = 8, then there exist disjoint subsets A, B of X both of cardinality 2 such that the sum of the elements in A is equal to the sum of the elements in B. Hint: Use the pigeonhole principle by defining a function that sends a subset to the sum of its elements.

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