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Predicate Logic

In predicate logic, we define functions on sets. Let \(S\) be a set. A predicate is a function \[P : S \rightarrow \{T, F\}\] For every \(x \in S,\) \(P(x)\) is a statement. When \(P(x) = T,\) \(x\) makes the predicate \(P\) true. When \(P(x) = F,\) \(x\) makes the predicate \(P\) false.

Example: Let \(S\) be the set of all people alive today. Let \(P\) be the predicate \(P(x) =\) "\(x\) lives in the United States."

As I am writing this, Donald Trump is an element of \(S,\) meaning he is alive. Also, \(P(\)Donald Trump\()=T\) because it is true that Donald Trump lives in the United States.

Similarly, it is true that J.K. Rowling is in \(S,\) but \(P(\)J.K. Rowling\()=F\) since she does not live in the United States.

Finally, Abraham Lincoln\(\not\in S\) since Abraham Lincoln is not currently alive. Therefore, \(P(\)Abraham Lincoln\()\) is undefined.

Define \(P(x)\) to be the predicate statement "\(x\) is an odd number." Which of the following is false?

Let \(P(x)\) be the predicate statement "I like \(x\)"" and let \(Q(x)\) be the predicate statement "I do not like \(x.\)" Which statement means "I like apples, but I do not like oranges."?