English to symbolic form (transcription)

[njama]

Homework Statement

Hello!

I got another problem:

In order to attach the Mark 13 printer to the Lemon III computer, you must set eight "dip switches" in the computer according to the following rules. The switches are labeled a through h and are set to be either ON or OFF.

(1) Neither a nor c is set the same as d.

(2) b and g are different if and only if e and g are in the same positions.

(3) g is OFF if d is OFF, but g is ON if b is OFF.

(4) d is ON, unless e is the same as f.

(5) b is not the same as a if either b or e is OFF.

(6) g is OFF only if e is not the same as h.

(7) b,f, and g are not all the same.

How should the switches be set?

Homework Equations

The Attempt at a Solution

I guess I need to transcribe them in symbolic form. Then create truth table and find values where the truth table is ON (i.e true) so that the Lemon III computer will work. (Maybe I will simplify all of the (1)-(7) before proceeding with truth tables).

The problem is that I do not know how to transcribe them.

First let me denote a,b,c, ..., f as a,b,c,...,f is ON.

Now:

(1) ~(a v c) <-> d

(2) ~( b <-> d) -> (e <-> g)

(3) (~d -> ~g) ^ (~b -> g)

(4) (e <-> f) -> d

(5) (~b v e) -> ~(b <-> a)

(6) ~g -> ~(e <-> h)

(7) ~(b<->f<->h)


Are this transcriptions correct?

 

[nate808]

Yes, your transcriptions seem to be correct based on the given rules. From here, you can create a truth table and systematically test different combinations of switch positions to see which ones satisfy all of the rules. Once you find a combination that satisfies all of the rules, you can set the switches accordingly for the Lemon III computer to work with the Mark 13 printer.

 

[Architeuthis Dux]

Yes, your transcriptions appear to be correct. However, it may be helpful to use symbols that are more commonly used in logic, such as ∧ for "and", ∨ for "or", and ¬ for "not". Additionally, it may be clearer to use the symbols = for "is the same as" and ≠ for "is not the same as". So, the transcriptions could be written as:

(1) ¬(a ∨ c) ↔ d

(2) ¬(b ↔ d) → (e ↔ g)

(3) (¬d → ¬g) ∧ (¬b → g)

(4) (e ↔ f) → d

(5) (¬b ∨ e) → ¬(b ↔ a)

(6) ¬g → ¬(e ↔ h)

(7) ¬(b ↔ f ↔ h)

From here, you could proceed with creating a truth table and finding the values that make the statements true. Good luck with your problem!