Set Theory: Why use implication?

[raddian]

Homework Statement

I hope this does not violate copyright or anything but this problem originated from an assignment from Introduction to Mathematical Thinking in Coursera. I could not post there because the class ended and the discussion board there is dead.

Let C be the set of all cars, let D(x) mean that x is domestic, and let M(x) meant that x is badly made. Express the following in symbolic form using these symbols:

(a) All domestic cars are badly made.

(The statement above is fictitious)

Homework Equations

The professor said that the answer is [strikeout]$$( \exists x \in C)[D(x) \implies M(x)]$$[/strikeout]

Corrected:
$$( \forall x \in C)[D(x) \implies M(x)]$$

The Attempt at a Solution

My answer

Corrected:
$$( \forall x \in C)[D(x) = M(x)] $$

[strikeout]$$( \exists x \in C)[D(x) = M(x)] $$[/strikeout]

I think it's D(x) = M(x) because all domestic cars are badly made.
"Are" is the plural for "is"; therefore, I think an equals (=) sign is necessary.

 

[willem2]

I don't think you copied the answer of your professor incorrectly.

You do not yet understand the meaning of some of the logical expressions.

D(x) means: x is a domestic car

D(x) = M(x) means: (x is a domestic car and x is badly made) OR (x is not a domestic car and x is not badly made). The whole statement will be satisfied if a badly made domestic car exists, or if a well-made foreign car exists. In the first case D(x) and M(x) are both true and in the second case they are both false.

What you want to express: is that if you pick a random car, and it is domestic, it's guaranteed to be badly made. This goes easier with the universal quantifier $\forall$.

 

[raddian]

Oh my bad, I will edit my post. I accidentally used $\exists$ when I should have used $\forall$.

 

[Fredrik]

I think it's D(x) = M(x) because all domestic cars are badly made.
"Are" is the plural for "is"; therefore, I think an equals (=) sign is necessary.

An equality sign doesn't make much sense there. It would mean that D(x) and M(x) represent exactly the same statement. So if D(x) represents (i.e. is a notation for) the statement "x is domestic." and D(x)=M(x), then M(x) would represent exactly the same string of text: "x is domestic.".

Did you mean to put an equivalence arrow there? The statement
$$\forall x\in C~D(x)\Leftrightarrow M(x)$$ is saying that every car is domestic if and only if it's badly made. So no foreign car is badly made. This is not what the original statement is saying.

 

[raddian]

An equality sign doesn't make much sense there. It would mean that D(x) and M(x) represent exactly the same statement. So if D(x) represents (i.e. is a notation for) the statement "x is domestic." and D(x)=M(x), then M(x) would represent exactly the same string of text: "x is domestic.".

Did you mean to put an equivalence arrow there? The statement
$$\forall x\in C~D(x)\Leftrightarrow M(x)$$ is saying that every car is domestic if and only if it's badly made. So no foreign car is badly made. This is not what the original statement is saying.

I really meant to put an = sign. I'm new to set theory. Like I thought it would be okay because $$(\forall x \in R)(x>10)$$, where R denotes the set of all real numbers (If anyone can explain how to display the standard hollow "R" using LaTeX; I used \mathBB{R} but it didnt work). I thought = worked the same way. Thank you both for your replies. I'm beginning to understand why the equal's sign is not appropriate.

 

[AlephZero]

You need to be clear about what the symbols mean. In your example $(\forall x \in \mathbb{R})(x > 10)$ $x$ is a real number, so $x > 10$ or $x = \pi/2$ mean what you expect.

Latex note: you want \mathbb{R}, not \mathBB{R}.

But $D(x)$ and $M(x)$ are functions of $x$ that have the value "true" or "false" for a particular value of $x$. If you replace them with the sort of functions you are familiar with, you wouldn't write something like $x^2 > 4 = |x| > 2$.

You could write "the set of x's such that x^2 > 4 is equal to the set of x's such that |x| > 2": $(x \in \mathbb{R})(x^2 > 4) = (x \in \mathbb{R})(|x| > 2).$

Or you could say "if x is a real number and x^2 > 4, then |x| > 2": $(\forall x \in \mathbb{R})(x^2 > 4 \implies |x| > 2 )$.

 

[verty]

An old symbol for implication was the less equal sign: D(x) <= M(x): if D(x) = 1, M(x) = 1. And equivalence used to use the equal sign: D(x) = M(x). But that style became obsolete about a century ago.

 

[Fredrik]

You could write "the set of x's such that x^2 > 4 is equal to the set of x's such that |x| > 2": $(x \in \mathbb{R})(x^2 > 4) = (x \in \mathbb{R})(|x| > 2).$

The notation should be
$$\big\{x\in\mathbb R\big|\,x^2>4\big\}=\big\{x\in\mathbb R\big|\,|x|>2\big\}$$ I would interpret the notation $(x \in \mathbb{R})(x^2 > 4) = (x \in \mathbb{R})(|x| > 2)$ as the false claim that
$$(x \in \mathbb{R})(x^2 > 4)$$ and
$$(x \in \mathbb{R})(|x| > 2)$$ are the same string of text.

 

[micromass]

The notation should be
$$\big\{x\in\mathbb R\big|\,x^2>4\big\}=\big\{x\in\mathbb R\big|\,|x|>2\big\}$$ I would interpret the notation $(x \in \mathbb{R})(x^2 > 4) = (x \in \mathbb{R})(|x| > 2)$ as the false claim that
$$(x \in \mathbb{R})(x^2 > 4)$$ and
$$(x \in \mathbb{R})(|x| > 2)$$ are the same string of text.

I wouldn't interpret that as such. I would just interpret it as a formula that isn't even a well-formed formula. In either sense, one shouldn't use equality here.

 

[Fredrik]

I really meant to put an = sign. I'm new to set theory. Like I thought it would be okay because $$(\forall x \in R)(x>10)$$, where R denotes the set of all real numbers (If anyone can explain how to display the standard hollow "R" using LaTeX; I used \mathBB{R} but it didnt work). I thought = worked the same way. Thank you both for your replies. I'm beginning to understand why the equal's sign is not appropriate.

You can certainly say things like $(\forall x\in\{-1,1\})(|x|=1)$, but in an example like this, the the left-hand side and right-hand side are both sets. (In ZFC set theory, there's nothing but sets. The elements of sets are themselves sets. Integers are sets. Real numbers are sets. Functions are sets. Even relations like the < relation on $\mathbb R$ are sets). But your D(x) and M(x) are not sets. They are strings of text in the language of set theory, which can be given truth values by a specification of what set the symbol x represents.

Your example says that all real numbers are greater than 10. This is fine unless you meant to write down a true statement.

 

[Fredrik]

I wouldn't interpret that as such. I would just interpret it as a formula that isn't even a well-formed formula.

Ah yes, that's much better.

 

[raddian]

I understand now guys. Thanks for the help (: