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2. A = Z, the set of integers with usual addition and multiplication. Prove if true or provide a counterexample if false for

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Answer #1
  • TRUE. If a+x = b, then , on adding -a to both the sides, x = b-a is the unique solution.
  • TRUE. The solution is x = y = b-a.
  • TRUE only if a, b ≠ 0, otherwise FALSE. Since there is no restriction on a and b, the statement is FALSE.
  • FALSE. Since there is no restriction on a and b, these can be 0. Then x and y are indeterminate. TRUE, if a, b ≠ 0.
  • TRUE only if a, c ≠ 0 and a ≠ c, otherwise FALSE. Then the solution is x = (d-b)/(a-c). Since there is no restriction on a and c, the statement is FALSE.
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