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Bob4040
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Is the set of all integers where <x,y> [itex]\in[/itex] R if and only if (y = x + 3) & (y = x – 3) an empty set?
An empty set is a set that has no elements. In other words, there are no objects or numbers contained within the set.
The phrase "if and only if" is a logical statement that means both conditions must be met in order for the statement to be true. In this case, both conditions of (y = x + 3) and (y = x – 3) must be true for the set to be empty.
This is a contradiction, as no number can be both 3 more and 3 less than the same number. Therefore, the statement cannot be true and the set is empty.
No, there is no possible set of numbers that would satisfy both conditions of (y = x + 3) and (y = x – 3) at the same time. Therefore, the statement is always false and the set is always empty.
This concept is related to the scientific method, where a hypothesis is only valid if it can be proven true under all conditions. In this case, the statement cannot be proven true and therefore, the set is always empty.