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# If at a step px is false then x px is false and the

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Unformatted text preview: as “For some x, P(x)”, or as “There is an x such that P(x)”, or “For at least one x, P(x)”. Examples: If P(x) denotes “x > 0” and domain is the integers, then ∃x P(x) is true. It is also true if domain is the positive integers. If P(x) denotes “x < 0” and domain is the positive integers, then ∃x P(x) is false. If P(x) denotes “x is even” and domain is the integers, then ∃x P(x) is true. Uniqueness Quan\$ﬁer ༉  ∃! x P(x) means that P(x) is true for one and only one x in the universe of discourse. ༉  This is commonly expressed in English with the following ways: ༉  “There is a unique x such that P(x)”. ༉  “There is one and only one x such that P(x)”. ༉  Examples: If P(x) denotes “x + 1 = 0” and domain is the integers, ∃! x P(x) is true. But if P(x) denotes “x > 0”, then ∃!x P(x) is false. Thinking about Quan\$ﬁers ༉  When the domain of discourse is ﬁnite,...
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