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Stream: deprecated: mathematics

Topic: Analyzing Logic of Statements

Jacob Zelko (May 25 2023 at 17:49):

Hi folks, making good progress with How To Prove It! I was asked an interesting problem that Ifound rather confusing which was the following:

Analyze the logical forms of the following statements:

1) 3 is a common divisor of 6, 9, and 15.

2) x and y are men, and either x is taller than y or y is taller than x.

For 1), I found this the harder of the two and tried this approach: I said to let $A$ represent the statement 3 is a divisor and $B$ represent 6 is divisible by 3, $C$ represent 9 is divisible by 3, and 15 is divisible by 3. Then, I said the logical form is $A \land (B \land C \land D)$. I didn't really like this answer as it feels contrived but I also didn't quite know how to approach this to make sure I capture the correct relationships.

For 2), I had a much easier time. I said let $M(x)$ represent x is a man and $T(a, b)$ represent a is taller than b. For the logical representation, I said, $(M(x) \land M(y)) \land (T(x, y) \lor T(y, x))$. I felt better about this answer as I'd read it back the same way as the original statement.

Does it seem like I am on the right track? Thanks all!

David Egolf (May 25 2023 at 19:02):

For (1), I'd wonder if it could be helpful to let $D(x,y)$ mean "$x$ is a divisor of $y$". By the way, I don't really understand what you mean by your $A$ "3 is a divisor".

Jacob Zelko (May 25 2023 at 20:50):

You know, I actually did try that but I found it additionally confusing. Let me share what I wrote in a moment @David Egolf

Jacob Zelko (May 25 2023 at 21:37):

Here is more of what I put:

$D(x)$ which represents x is divisible by 3. So then I could say the statement: $D(6) \land D(9) \land D(15)$. The only problem is is that I get confused by how I should capture the statement that 3 is the common divisor of these statements unless I am doing so implicitly with how I formulated the statement. Any thoughts?

David Egolf (May 25 2023 at 21:47):

In words, $D(6) \land D(9) \land D(15)$ says "6 is divisible by 3, and 9 is divisible by 3, and 15 is divisible by 3". I think that's what it means to say that 3 is a common divisor of 6, 9, and 15.

Jacob Zelko (May 25 2023 at 21:57):

Ah cool! That's what I was initially trying but it just felt so clunky. Thanks for the sanity check/second opinion here @David Egolf !

Spencer Breiner (May 25 2023 at 22:27):

If you wanted to emphasize the commonality, you could use an existential quantifier:
$\exists x. (x|6) \wedge (x|9) \wedge (x|15) \wedge (x=3)$

Spencer Breiner (May 25 2023 at 22:28):

The formulas are equivalent, but this is arguably a bit closer to the intensional meaning of the sentence.