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21

"A only if B" means that you can't have A without B, i.e. $\neg(A\wedge(\neg B))$, which simplifies (via de Morgan) to $(\neg A)\vee B$.

13

No, you cannot depend on that. If it were that simple, we wouldn't need clunky phrases like "exclusive or" to make clear when an "or" is exclusive. Linguistically, "either" is simply a marker that warns you in advance that an "or" is going to follow. Nothing more.

12

What you have encountered is a variation on vacuous truth. If the premise is false, then the statement is true. For instance, I can say "If I'm from Mars, then it will rain candy tomorrow", and that will be a true statement. It is true because it is not a lie. It could only be a lie if I really were from Mars, and at the same time it didn't rain candy ...

8

No, we cannot omit the existential quantifier. How we read the formula: $x=2m$ ? As: "every $x$ is even" ? Or as: "some $x$ is even" ? Or as: "every $x$ is the double of every $m$" ? Or as: "some $x$ is the double of every $m$" ? The "usual" convention is that we can omit the universal quantifier. If we apply it, we read: $\exists m \ (x = 2m)$ as:...

7

Consider a world with no people. Then there is no comedian, so your translation is correct. However, the statement is false because there doesn't exist anyone, in particular no one who satisfies $C(x) \rightarrow F(x)$. To correct your answer, you could say that in case 1, there is someone who is not a comedian.

6

"A if B". When B is true then A must be true (sufficient). "A only if B". When B is false then A must be false (necessary). "A if and only if B". When B is true A must be true, When B is false A must be false. (necessary and sufficient) "if" allows A to be any truth value when B is false. "only if" allows A to be any truth value when B is true.

6

The two answers are equivalent. "$\lnot \forall$" is the same as "$\exists \lnot$". If not all cats are black, there must be some cat that is not black. Thus, we have that $$¬∀x \ (Gx \to Lxx) \iff ∃x \ ¬(Gx \to Lxx) \text{.}$$ Now we apply the tautological equivalence $$\lnot (p \to q) \iff (p \land \lnot q)$$ (We can check it with a truth-table:...

6

Just because the monkey is home, that doesn't automatically mean that Tarzan is happy. That's how "only if" is interpreted in conventional mathematical English. Tarzan can't be happy if the monkey is away, but when the monkey is there, Tarzan could be either happy or non-happy.

6

No, the second claim should be $r \to p$: $f$ being integrable $p$ is a necessary condition, rather than a sufficient condition. And when something $p$ is a necessary condition for something else $q$, you translate it as $q \to p$ Also, I much prefer to present these statements as three separate statements as part of an argument, rather than as one big ...

6

First-order logic does not have a notion of "many" - this, like "most," "almost all," etc. is an example of a generalized quantifier, and handling them takes us to extensions of first-order logic. We can express each of the following in first-order logic: "Some printer is broken." ($\exists x(p(x)\wedge b(x))$) "Multiple printers are broken." (\exists x,y(... 6 A correct formalization of the phrase "an integer is even iff it equals double some other integer" is the following: \begin{align} \forall x \in \mathbb{Z} \, (\exists y \in \mathbb{Z} \, (x = 2y) \iff \textrm{even}(x)) \end{align} 5 Considering "unless" means "without", "if not", we have several ways to translate "P$unless/except$R$". Note that Without$R$,$P$is true. We can write "$\sim R\to P$". Either$P$or$R$is true. We can write "$P$or$R$" If$P$fails,$R$is the reason. We can write "$\sim P\to R$" Moreover, "$P\to Q$" is logically equivalent to "$\sim P$or$Q$", ... 5 Interpreting "A when B" as$A \Rightarrow B$isn't really consistent with the english language. If I say "I get wet when it rains", that means (at least to me) that rain implies getting wet, not the other way around. I.e., whenever it rains I get wet, but I might get wet even if it doesn't rain (if I, say, jump into the pool). I'd thus read "A when B" as$B \...

5

Here is a table showing all possible value-combinations for A and B in the first two columns. You can see that columns 3 and 5 are the same, therefore both predicates must be equivalent. | A | B | A->B | not A | (not A) or B | |---|---|------|-------|--------------| | T | T | T | F | T | | F | T | T | T | T | | T | F ...

5

Notice how '$x$ is even' makes explicit reference to $x$, but not to $m$. That should be an indication already that in your formula you want $x$ as a free variable (i.e do not quantify the $x$) but not $m$, so use: $$\exists m \ x = 2m$$ Also note that depending on how you quantify the $x$, you get a different statement: $$\forall x \exists m \ x =2m$$ ...

5

A) Wrong. Step by step we find that the following are negations of $\forall x\exists y\neg P(x,y)$ $\neg\forall x\exists y\neg P(x,y)$ $\exists x\neg\exists y\neg P(x,y)$ $\exists x\forall y\neg\neg P(x,y)$ $\exists x\forall yP(x,y)$ In the last one $\neg$ is not present. B) The original statement is true. For every $x\in\mathbb R-\{0\}$ we can find ...

4

"What are the rules of omission of parentheses?" The answer is: different rules in different texts! But what is really being asked is something different, i.e. "What are the rules for using dots instead of parentheses?" [and I have edited the title question to fit]. Again, though, the answer is that there are different rules in different texts. Church's ...

4

I will follow the suggestion of S.C.Kleene, Mathematical Logic (1967), where he summarize in a table a list of expressions and their possible translation in symbols (pag.64) : $A \lor B$ is $A$ unless $B$ [usually] and is $A$ except when $B$ [usually]. Try now with the exercises : Exercise 3 : If $P$, $Q$ and $R$ are translations for “$x=y$” , “$x/z=y/z$...

4

A straightforward translation is there are an object $y$ and a person $x$ such that $x$ does not own $y$. This still sounds more like mathematics than like everyday English, so we try to improve it. That there is someone who does not own a particular object means precisely then it’s not the case that everyone owns that object, so we can further translate:...

4

Most introductory books on axiomatic set theory directly define set theoretical concepts in the language of first order logic. Once you know these definitions, all you need to do is translate, translate and translate until you are satisfied. One such book that I use can be found here: http://www.amazon.com/Theory-Continuum-Problem-Dover-Mathematics/dp/...

4

Let's actually make a truth table for "even if". We get: \begin{array}{cc|ccc} P&Q&P&\text{ even if }&Q\\\hline T&T&T&\mathbf{T}&T\\ T&F&T&\mathbf{T}&F\\ F&T&F&\mathbf{F}&T\\ F&F&F&\mathbf{F}&F \end{array} where the first two are trivial, and $P$ being false and $Q$ being true ...

4

In everyday speech, "or" is usually exclusive even without "either." In mathematics or logic though "or" is inclusive unless explicitly specified otherwise, even with "either." This is not a fundamental law of the universe, it is simply a virtually universal convention in these subjects. The reason is that inclusive "or" is vastly more common.

4

Why is “A only if B” equivalent to “(not A) or B”? Let's make it less abstract: A = You can have your pudding B = You eat your meat So "A only if B" means "You can have your pudding only if you eat your meat". How can you avoid having pudding but not eating meat? Either by not having pudding or by eating your meat. Hence $\neg A \lor B$. It's my ...

4

a,b,c are correct. Issues with d. First, odd and even are switched ($x|y$ means $x$ divides evenly into $y$, not the other way around). Second, more importantly, it should be $\exists x\exists y(even(x)\wedge odd(y) \wedge y|x)$ ($odd(x)|even(y)$ has no meaning). Same two issues with e. You can't use $1|prime(x)$ to mean "x is prime and 1 divides x". For ...

4

No. A paradox doesn't just assert something incorrect (e.g. "$0\not=0$") - it asserts something which cannot be consistently assigned a truth value. Just implying the negation of a tautology doesn't mean that a statement is paradoxical: e.g. "$p$ and $\neg p$" is not a paradox, it's just a false statement.

4

$(n\in A)\to P(n)$ is a good start, but you need something to express the idea that you want this to be true for all $n$. So you should write $$\forall n : \bigl[ n\in A \to P(n) \bigr]$$ This is what "$\forall n\in A. P(n)$" (with variations in punctuation that don't carry any meaning) is usually considered to be an abbreviation for in formal logic.

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