first order logic notations

Question

can someone please tell when to interpret this symbol $\Leftrightarrow$ as logical equivalence and when as double implication?

Comments

We write f1=>f2 iff f1->f2 is taulogy.

If above follows then  implication becomes logical implication.

Answer 1

Bidirection ≡ Ex-Nor ≡ iff ≡ Equivalence operator ≡ <=>

p<=>q means   (p=>q )AND(q>p) which means (p'+q).(q'+p) which means p'q'+pq

Now the prepositional logic  p'q'+pq would be true when either both p,q are false or both are true.so this make p,q logically equivalent iff p'q'+pq is true always.

like for example let p=a+b' and q= (a'b)' so as you can guess p is logically equivalent to q here

but you can prove it by using p<=>q if the outcome for all possible inputs is always true then p and q would be logically equivalent.

Comments

I think you can't say p<=>q is logically equivalent when output of all the input isTRUE.It must be like proving LHS=RHS and it may be F<=>F correct me if am wrong !

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Hello junaid.

you didn't get what i said.

I said let the two prepositional logic formula p and q now we have to tell whether both formulas are logically equivalent or not. how can we ?

if we can prove p<=>q is true for all possible outcomes for all possible inputs of p and q then they are logically equivalent.

If the outcome of p<=>q is false that mean out of those p and q one is false and one is true but when one is true for some input and other is false for same input , how can be they logically equivalent .

logically equivalent mean for same input they should generate same output.

p<=>q formula is used to prove logical equivalence because its output can only be true when both p and q are same (both true or both false).

here your LHS=p and RHS =q

now for LHS=RHS if for same input they generate same output

or otherwise LHS= p<=>q  , RHS is T , LHS=RHS when p and q are logically equivalent.

hope you get it now ?

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I think this condition is for bi-conditional statement not for logical equivalence.

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i'm really not getting , what are you not getting. please tell me more clearly.

i am trying one more time.

If for same input p and q are generating different then their bidirectional formula (p<=>q) would generate output F that's simply mean p and q are not logically equivalent.

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Compound prepositions  p and q are called logically equivalent if p<=>q is a tautology.

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I get what you trying to say but i am just saying that p<->q is not the same thing as p<=>q.

and p<=>q is logically equivalent if p logically implies q and q logically implies p.

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Bhai.

p<=>q or p<->q both are same thing , they are notations brother like p=>q or p->q

i said naa p<=>q means p iff q .

p<=>q is logically equivalent is technically wrong statement , here we are talking about the equivalence of p and q. take any to logic statements in this world , any two.

taken ?

good , now make them operators for EX-NOR operation if the outcome is T then your both taken logic formulas are equivalent otherwise not .

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hello rupendra thanx for ur answer.

i alread knew that p<=>q is logical equivalence symbol ie when p<->q is a tautology.

but my question was sometimes in question they give => which is the symbol of logical implication but in question it means implication and same is with <=> so how to know that they are talking about logical equivalence or bi implication.

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Hello vineet.

I didn't get your point. please elaborate it with some example. where you're confuse.