Let F(x, y) be the statement " x can fool y " where the...

Let $F(x, y)$ be the statement " $x$ can fool $y$ " where the domain consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody can fool Fred. b) Evelyn can fool everybody. c) Everybody can fool somebody. d) There is no one who can fool everybody. e) Everyone can be fooled by somebody. f) No one can fool both Fred and Jerry. g) Nancy can fool exactly two people. h) There is exactly one person whom everybody can fool. i) No one can fool himself or herself. j) There is someone who can fool exactly one person besides himself or herself.

Let $F(x, y)$ be the statement " $x$ can fool $y$ " where the domain consists of all people in the world. Use quantifiers to express each of these statements.
a) Everybody can fool Fred.
b) Evelyn can fool everybody.
c) Everybody can fool somebody.
d) There is no one who can fool everybody.
e) Everyone can be fooled by somebody.
f) No one can fool both Fred and Jerry.
g) Nancy can fool exactly two people.
h) There is exactly one person whom everybody can fool.
i) No one can fool himself or herself.
j) There is someone who can fool exactly one person besides himself or herself.

Key Concepts

Uniqueness and Counting

When statements refer to exact counts, such as 'exactly one' or 'exactly two', expressing these conditions requires a combination of quantifiers, equality, and additional constraints. This concept is critical for accurately translating statements that impose numeric restrictions on the individuals satisfying a predicate.

Identity and Inequality

This concept involves the use of the equality operator (=) and its negation (?) in logical expressions to distinguish between different individuals. It is especially relevant for statements that exclude self-referential cases or require that certain properties hold for distinct individuals.

Logical Connectives

Logical connectives like conjunction (?), disjunction (?), implication (?), and negation (¬) are used to combine and modify simpler statements into more complex formulas. They help form precise logical expressions that match the intended meaning of compound statements.

Domain of Discourse

This concept involves the set of all objects or individuals under consideration—in this case, all people in the world. Defining the domain is a necessary first step in formal logic, as it determines what the variables in the logical expressions represent.

Predicate Logic

Predicate logic is a framework for representing statements involving objects and their properties or relations. In this context, the binary predicate F(x, y) is used to denote the relation 'x can fool y,' allowing the construction of expressions that describe various fooling scenarios among people.

Quantifiers

Quantifiers such as the universal quantifier (?) and the existential quantifier (?) are fundamental to expressing statements in predicate logic. They enable the formulation of assertions that apply to every element in the domain or to the existence of some elements, respectively, which is key for translating natural language statements into logical expressions.

Transcript

00:01 All right, so we're going to blast through a couple logical quantifiers here.

00:05 If f of x y means that x can fool y, then going through here, everyone can fool fred.

00:10 That means that f of x fred is true for every value of x.

00:17 Let's make this a little bit smaller here.

00:19 F of x fred is true for all values of x, so we'll just do for all x right there.

00:32 That's our a.

00:34 B, evelyn can fool everybody.

00:36 That means we have fool evelyn.

00:38 X is true again for every x, so for all x f of evelyn x, evelyn can fool x.

00:49 What you see down here, everyone can fool somebody, so that means everyone is going to be for all x.

00:57 Everyone there exists some person y, such that x can fool y.

01:05 It's the conventional way to interpret that.

01:11 This could also in english mean that there is somebody that everyone can fool.

01:16 We just switch the quantifiers around.

01:19 There exists a y such that for all x, x can fool y.

01:25 This means that everybody has someone they can fool, and this one means that there is someone that everyone can fool.

01:33 This statement in english is ambiguous.

01:36 It almost certainly means the first one.

01:38 Anyway, for d, there is no one who can fool everyone.

01:41 It means that there does not exist an x such that for all y, x can fool y.

01:51 No one can fool everyone means that there is not someone who can fool everyone.

01:55 That is just this statement here is that there does some there is someone who can fool everyone, and we're negating it because there is not someone who can fool everyone.

02:05 For e, everyone can be fooled by someone, so everyone is x can be fooled by someone, which means there exists someone who can fool them.

02:15 For all x, there exists a y where y can fool x.

02:22 Again, this is a little bit this is ambiguous in english.

02:28 This could be that there exists someone, there exists a y who can for all x fool them, but again it is probably meant to be indicating this solution here.

02:45 For f, we're going to do here on the right, no one can fool both fred and jerry.

02:52 No one means there does not exist someone, there does not exist an x who can fool both fred and jerry.

03:02 So that's just because they can fool both fred and jerry...

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