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Exercises. Find the negations of the following propositions. (a) p: "2+2 = 4“ (b) q: "1 = 0“ (c) r: "Diamonds are a pearl's best friend.“ (d) s: "All the politicians in this town are crooks.“ Solution (a) ~p is the statement "it is not true that 2+2 = 4," or more simply, ~p: "2+2 4."
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Exercises Find the negations of the following propositions. (a)p: "2+2 = 4“ (b)q: "1 = 0“ (c)r: "Diamonds are a pearl's best friend.“ (d)s: "All the politicians in this town are crooks.“ Solution (a) ~p is the statement "it is not true that 2+2 = 4," or more simply, ~p: "2+2 4." (b) ~q: "1 0." (c) ~r: "Diamonds are not a pearl's best friend." (d) ~s: " Not all the politicians in this town are crooks."
Before we go on... • Notice that ~p is false, because p is true. However, ~q is true, because q is false. A statement of the form ~q can very well be true; it is a common mistake to think it must be false. • To say that diamonds are not a pearl's best friend is not to say that diamonds are a pearl's worst enemy. The negation is not the polar opposite, but whatever would deny the truth of the original statement. Similarly, saying that not all politicians are crooks is not the same as saying that no politicians are crooks, but is the same as saying that some politicians are not crooks. Negations of statements involving the quantifiers "all" or "some" are tricky. We'll study quantifiers in more depth when we discuss the predicate calculus.
More exercises… • p: "There is life on Mars.“ ~p: … • p: "All lacrosse players are tall.“ ~p:… • p: "My computer has some new software.“ ~p:…
Which of the following are statements? Comment on the truth values of all the statements you encounter; if a sentence fails to be a statement, explain why. 1. All swans are white. 2. The fat cat sat on the mat. 3. Look in thy glass and tell whose face thou viewest. 4. My glass shall not persuade me I am old. 5. Father Nikolsky penned his dying confession to Patriarch Arsen III Charnoyevich of Peç in the pitch dark, somewhere in Poland, using a mixture of gunpowder and saliva, and a quick Cyrillic hand, while the innkeeper's wife scolded and cursed him through the bolted door.
6. 1,000,000,000 is the largest number. 7. There is no largest number. 8. There may or may not be a largest number. 9. Intelligent life abouds in the universe. 10. This definitely is a statement. 11. The speaker is lying. 12. This is exercise number 12.
14. "potato" is spelled p-o-t-a-t-o-e. Let p: "Our mayor is trustworthy," q: "Our mayor is a good speller," r = "Our mayor is a patriot." Express each of the following statements in logical form 15. Although our mayor is not trustworthy, he is a good speller. 16. Either our mayor is trustworthy, or he is a good speller. 17. Our mayor is a trustworthy patriot who spells well. 18. While our mayor is both trustworthy and patriotic, he is not a good speller.