Simple Logic Question of Converting or to and

[Hypnos_16]

Simple Logic Question of Converting "or" to "and"

Homework Statement

As the title would have you believe it's really just a basic question.
I have to write (¬q <---> r) /\ (¬p \/ ¬r) ---> (p \/ ¬q) just using "¬'s" and "/\'s" i know how to change everything else, but i cannot, for the life of me remember the equivalent of (a \/ b) in terms of nots and and gates.

Homework Equations

n/a

The Attempt at a Solution

Someone please help,

 

[gneill]

I find it's useful to use a Karnaugh map in these cases, since it let's you spot the simplifications and the complements visually.

That being said, I also find it useful to remember that in digital circuits a NAND gate is equivalent to an OR gate with both inputs negated, and that a NOR gate is equivalent to an AND gate with both inputs negated. That is to say, simply move the 'NOT' circles from the front or back of the gate and change from AND or OR or vice-versa.

 

[Hypnos_16]

But it simple terms of symbols like "∧ and" "¬ not." What would be an equivalent for "or?" The Question i was given is.

Rewrite the following formula using only ¬ and ∧ (that is, write an equivalent formula which does not have ∨,→,↔). Start by saying how you express (p∨q),(p→q)and(p↔q).

(¬q↔r)∧(¬p∨¬r)→(p∨¬q)

With that being the formula i need to convert. I just don't recall how exactly to change ∨ to something with ∧'s and ¬'s

 

[gneill]

But it simple terms of symbols like "∧ and" "¬ not." What would be an equivalent for "or?" The Question i was given is.

Rewrite the following formula using only ¬ and ∧ (that is, write an equivalent formula which does not have ∨,→,↔). Start by saying how you express (p∨q),(p→q)and(p↔q).

(¬q↔r)∧(¬p∨¬r)→(p∨¬q)

With that being the formula i need to convert. I just don't recall how exactly to change ∨ to something with ∧'s and ¬'s

I suggested two tools that would allow you to "remember" how to convert ∨ expressions to ∧ expressions. Here they are converting A ∨ B to an equivalent using ∧:

 

[DeeNos]

i cant even find anything on googleI understand the importance of clear and precise language in logical statements. In this case, it seems like you are trying to convert a logical statement from using "or" to using "and." This can be achieved by using De Morgan's laws, which state that the negation of a disjunction (a statement using "or") is equivalent to the conjunction (a statement using "and") of the negations of the individual statements.

In your case, you can use De Morgan's laws to rewrite the statement as follows:

(¬q <---> r) /\ (¬p \/ ¬r) ---> (p \/ ¬q)

¬(q \/ ¬r) /\ (¬(¬p) /\ ¬¬r) ---> (p \/ ¬q)

(¬q /\ r) /\ (p /\ r) ---> (p \/ ¬q)

This is the equivalent statement using only "nots" and "ands." I hope this helps with your homework!