Let x, y, z be integers. Decide if each of the following statements is true orfalse. If the statement is true, then prove it. If the statement is false, then give a counterexample(a) If x(y − z) = 0, then y = z.
Question
Let x, y, z be integers. Decide if each of the following statements is true orfalse. If the statement is true, then prove it. If the statement is false, then give a counterexample(a) If x(y − z) = 0, then y = z.
Solution
The statement "If x(y - z) = 0, then y = z" is not necessarily true.
Proof: The equation x(y - z) = 0 can be true under two conditions:
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If x = 0, then regardless of whether y = z or not, the equation will hold true. For example, let x = 0, y = 1, and z = 2. Here, y is not equal to z, but x(y - z) = 0 * (1 - 2) = 0, which is true.
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If y = z, then regardless of the value of x, the equation will hold true. For example, let x = 2, y = 3, and z = 3. Here, y is equal to z, and x(y - z) = 2 * (3 - 3) = 0, which is true.
So, the statement "If x(y - z) = 0, then y = z" is false because there is another condition (x = 0) under which the equation can be true.
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