Understanding Logic Equivalence Laws

In this video, we are going to discuss about equivalence formulas or laws of logic. To check whether two prepositions are equivalent or not, we use two approaches. The first approach is using the truth table. The second approach is without using truth table. So without using truth table, with the help of the equivalence formulas, we can check whether two compound prepositions are equivalent or not so first let's see about equivalence formulas or can also be called as loss of logic totally 10 laws are there let us see all the 10 laws one by one the first law is the first law is idempotent law idempotent law in each law we will be having two types first one is based upon the disjunction that is r operator second lie is based upon the conjunction means under operator so first let's see what is idempotent law if we perform our operation or under operation on a single statement or a single preposition we will get the same proposition variable so here prp is logically equivalent to p why because let us assume that p is true value if p is true value then true p means true so true or true means true only so true is nothing but p value let us assume that p is false or here p is false so false or false means false only so false is nothing but p so if the problem contains a compound preposition like prp then we can replace that preposition with p so p r p is logically equivalent to p so this is the first formula and the second formula is dissension this conjunction conjunction p and p is equivalent to p and p is equivalent to p let p is true so true and true means true so true is nothing but p only let p is false so false and false means false so false is nothing but p value only so every law contains two rules okay the first one is based upon r r means what dissension the second line is based upon the conjunction and now let us see the second one the second law is identity law the second law is identity law here we have to remember one point identity law means we have to perform our operation with F. P or F. P or F is equal to, logically equivalent to, let P is true. So true or false means true. So true is nothing but the value of the P. Let P is false. Then false or false means false. False is nothing but the value of the P. So P or F is logically equivalent to P, whereas the second one is P and In place of f, we have to write true. True. So if you remember this one, then writing the remaining three rules will become quite easy. So p or f. Okay. The next one is conjunction p and f is replaced with t. So p and true is equal to. Let p is true. True and true means true. So true means the value of the p. Let p is false. False and true means false. False is nothing but the value of the p. So, p and true is logically equivalent to p. Now, let us see the third one. That third rule is dominance law. The third one is dominance law. So, what is dominance law? The first rule is p are. In place of f, we have to write true. In place of false, we have to write true. So, p are true is equal to. Let p is true. true or true means true let p is false false are true let p is false false or true means true so every time we are getting true only okay true or true or true or false means true only okay next the second rule is p and in place of true we have to write false in place of this true we have to write false p and false is equal to let p is true true and false means false Let p is false. So false and true means? false and false means false only so here what is the result false okay so why this is called as dominance law if we observe here here true and false are dominating when compared with the proposition or statement okay if we observe the previous m if we observe the identity law we got p as the result that proposition or statement as the result whereas if we observe this one here we are getting true and false as the result that means true and false are more dominant here when compared with the preposition so that's why this is called as dominance law next rule is associative law next rule is associative law let's see what is associative law associative if you observe here after a here we have s is there yes means same symbol yes means same symbol here this association is applied on three prepositions pr q r r so what is this yes this yes means same symbol here if a problem contains pr f then it should be replaced with p if a problem contains p and true it is replaced with p if a problem contains p or t then it is replaced with t if a problem contains p and f it is replaced with f these are called as equivalent ross p are true is logically equivalent to true okay next associative means if you observe the second letter yes yes means same symbol here we have r here we have r within the expression also we have r after the p also we have r that means the same symbol if if a problem is like this p r q r r then it is logically equivalent to p r q r r p r q r r so what is the second one same symbol in place of r we have to take under symbol so p and q and r is logically equivalent to p and q and r p and q and r so if a problem contains p and within the parenthesis q and r we can replace that with p and q and r or if the problem contains p and q within the parenthesis and r then we can replace this rhs with this lhs also okay now let us see the next one after associative the next law is distributive law here d stands for different here this is not the abbreviation for your understanding purpose i am giving some meaning yes means same this is not the abbreviation okay for your understanding purpose same symbols within the parentheses same symbol and before the parentheses also same symbol okay Distribute means different symbol. Different symbol. That means here we have after P we have R. Then within the expression we will have different symbol. So this D means different. P R Q and R is logically equivalent to P R Q. It is logically equivalent to P R Q and P R R. P R Q. P R Q and. prr but we should not expand this rule like this we should not like we should not write like prq or prr prq or prr that is uh we don't have any rule like that okay associative means we can write like this only this is called as distributive different are simple and simple so prq and prr okay let us take the second one p and different symbol p and q r r so before the left parenthesis we have under symbol so we should write a different symbol within the expression so p and q r r is logically equivalent to p and q r p and q r p and r p and q r p and r okay now let us write the next rule the next rule is commutative law commutative law here commutative law is the simplest law among all the laws commutative means very simple prq is logically equivalent to qrp why because what is our operation here it doesn't depends upon the order okay when both are true when both the statements are true the result is true otherwise the result is this is our operator what is our operator when both the statements are false the result is false otherwise the result is true only it doesn't depends upon the order if you take p implies q p implies q depends upon the order When p is true, q is false, the result is false. Whereas, p, r, q or p and q doesn't depend upon the order. When both are true, here, r means when both are false, the result is false. Otherwise, the result is true. If a problem contains p, r, q, then we can replace with q, r, p also. p, r, q is logically equivalent to q, r, p. So, the second rule is also simple. p and q is logically equivalent to q and p why because we know about p and q if both are true the result is true otherwise the result is false okay so we can substitute p and q with q and p so when q and p are true the result is true otherwise the result is false okay let us take the next law that is absorption law that is absorption law absorption law already we have seen about association law association law means uh both must be same symbol whereas absorption law means here the both must be different symbols okay whereas in association law we require three statements whereas in absorption law we require only two statements pq two statements are enough okay so pr association means same whereas we have one more called absorption in absorption we have to take different Different. PR, PRQ. Association means we require three statements, three propositions. Variance absorption means two are sufficient. So PR, P and Q is logically equivalent to P. Is logically equivalent to P. If you want, just you can substitute true value and false value. So you will get the result. And what is the second one? P and here they are different. Symbols are different. So P and PRQ. P and and R. They are different. it is logically equivalent to 2p it is logically equivalent to 2p so if a problem contains like this we can substitute with p or if the problem contains like this also we can substitute with the p okay now let us see the next one the next one is d mogran's law d morgan's law so this is very very popular law this is very very popular law the first one is negation of prq negation of prq implies Negation P. Negation P. Negation of R means and. Negation of R means and. And negation Q. Negation Q. B. So if the problem contains negation of P or Q then we can replace with negation P and negation Q. Or if a problem contains negation P and negation Q we can replace with negation P or Q. According to our problem requirement we have to substitute. RHS in place of LHS or LHS in place of RHS. And the second one is negation of. This column represents what? Conjunction and whereas this column represents disjunction R. So negation of P and Q. Negation of P and Q is logically equivalent to negation P or negation Q. Negation P or negation Q. If a problem contains negation of P and Q you can replace with negation P or negation Q. or if a problem contains negation p or negation q you can replace with negation of p and q now let us see the next one the next one is negation law negation or there are several names are there we can call this as negation or complement complement negation or complement law let us see the first one negation if you take p then what is the negation negation p p or negation p if you take p then its negation is negation p and this is nothing but what r loss disjunction so what is the result of p or negation p if you take p as true true or negation p means false true or false means true if you take p as false false or negation false means true so it always returns true as the result and the second law is p and negation p is equivalent to if you take p as true true and negation true true one negation true means false true and false means false if you take p as false false and negation false means true false and true means false we always get false as the result and the last last law is double negation double negation law double negation law negation of negation p is equivalent to negation of negation p is equivalent to p if you take p as true if you take p as true negation true means what false negation of false means true so the true is nothing but p value if you take p as false negation false means true negation of true means false so false is nothing but p value so by using these 10 rules or 10 formulas we can check whether two propositions are equivalent or not without using the truth table so these are the 10 formulas in the next video we'll see various examples regarding this approach.

So without using truth table, with the help of the equivalence formulas, we can check whether two compound prepositions are equivalent or not so first let's see about equivalence formulas or can also be called as loss of logic totally 10 laws are there let us see all the 10 laws one by one the first law is the first law is idempotent law idempotent law in each law we will be having two types first one is based upon the disjunction that is r operator second lie is based upon the conjunction means under operator so first let's see what is idempotent law if we perform our operation or under operation on a single statement or a single preposition we will get the same proposition variable so here prp is logically equivalent to p why because let us assume that p is true value if p is true value then true p means true so true or true means true only so true is nothing but p value let us assume that p is false or here p is false so false or false means false only so false is nothing but p so if the problem contains a compound preposition like prp then we can replace that preposition with p so p r p is logically equivalent to p so this is the first formula and the second formula is dissension this conjunction conjunction p and p is equivalent to p and p is equivalent to p let p is true so true and true means true so true is nothing but p only let p is false so false and false means false so false is nothing but p value only so every law contains two rules okay the first one is based upon r r means what dissension the second line is based upon the conjunction and now let us see the second one the second law is identity law the second law is identity law here we have to remember one point identity law means we have to perform our operation with F. P or F. P or F is equal to, logically equivalent to, let P is true.

So true or false means true. So true is nothing but the value of the P. Let P is false.

Then false or false means false. False is nothing but the value of the P. So P or F is logically equivalent to P, whereas the second one is P and In place of f, we have to write true.

True. So if you remember this one, then writing the remaining three rules will become quite easy. So p or f. Okay.

The next one is conjunction p and f is replaced with t. So p and true is equal to. Let p is true.

True and true means true. So true means the value of the p. Let p is false.

False and true means false. False is nothing but the value of the p. So, p and true is logically equivalent to p. Now, let us see the third one.

That third rule is dominance law. The third one is dominance law. So, what is dominance law? The first rule is p are. In place of f, we have to write true.

In place of false, we have to write true. So, p are true is equal to. Let p is true. true or true means true let p is false false are true let p is false false or true means true so every time we are getting true only okay true or true or true or false means true only okay next the second rule is p and in place of true we have to write false in place of this true we have to write false p and false is equal to let p is true true and false means false Let p is false.

So false and true means? false and false means false only so here what is the result false okay so why this is called as dominance law if we observe here here true and false are dominating when compared with the proposition or statement okay if we observe the previous m if we observe the identity law we got p as the result that proposition or statement as the result whereas if we observe this one here we are getting true and false as the result that means true and false are more dominant here when compared with the preposition so that's why this is called as dominance law next rule is associative law next rule is associative law let's see what is associative law associative if you observe here after a here we have s is there yes means same symbol yes means same symbol here this association is applied on three prepositions pr q r r so what is this yes this yes means same symbol here if a problem contains pr f then it should be replaced with p if a problem contains p and true it is replaced with p if a problem contains p or t then it is replaced with t if a problem contains p and f it is replaced with f these are called as equivalent ross p are true is logically equivalent to true okay next associative means if you observe the second letter yes yes means same symbol here we have r here we have r within the expression also we have r after the p also we have r that means the same symbol if if a problem is like this p r q r r then it is logically equivalent to p r q r r p r q r r so what is the second one same symbol in place of r we have to take under symbol so p and q and r is logically equivalent to p and q and r p and q and r so if a problem contains p and within the parenthesis q and r we can replace that with p and q and r or if the problem contains p and q within the parenthesis and r then we can replace this rhs with this lhs also okay now let us see the next one after associative the next law is distributive law here d stands for different here this is not the abbreviation for your understanding purpose i am giving some meaning yes means same this is not the abbreviation okay for your understanding purpose same symbols within the parentheses same symbol and before the parentheses also same symbol okay Distribute means different symbol. Different symbol. That means here we have after P we have R. Then within the expression we will have different symbol.

So this D means different. P R Q and R is logically equivalent to P R Q. It is logically equivalent to P R Q and P R R. P R Q. P R Q and. prr but we should not expand this rule like this we should not like we should not write like prq or prr prq or prr that is uh we don't have any rule like that okay associative means we can write like this only this is called as distributive different are simple and simple so prq and prr okay let us take the second one p and different symbol p and q r r so before the left parenthesis we have under symbol so we should write a different symbol within the expression so p and q r r is logically equivalent to p and q r p and q r p and r p and q r p and r okay now let us write the next rule the next rule is commutative law commutative law here commutative law is the simplest law among all the laws commutative means very simple prq is logically equivalent to qrp why because what is our operation here it doesn't depends upon the order okay when both are true when both the statements are true the result is true otherwise the result is this is our operator what is our operator when both the statements are false the result is false otherwise the result is true only it doesn't depends upon the order if you take p implies q p implies q depends upon the order When p is true, q is false, the result is false. Whereas, p, r, q or p and q doesn't depend upon the order. When both are true, here, r means when both are false, the result is false.

Otherwise, the result is true. If a problem contains p, r, q, then we can replace with q, r, p also. p, r, q is logically equivalent to q, r, p. So, the second rule is also simple. p and q is logically equivalent to q and p why because we know about p and q if both are true the result is true otherwise the result is false okay so we can substitute p and q with q and p so when q and p are true the result is true otherwise the result is false okay let us take the next law that is absorption law that is absorption law absorption law already we have seen about association law association law means uh both must be same symbol whereas absorption law means here the both must be different symbols okay whereas in association law we require three statements whereas in absorption law we require only two statements pq two statements are enough okay so pr association means same whereas we have one more called absorption in absorption we have to take different Different.

PR, PRQ. Association means we require three statements, three propositions. Variance absorption means two are sufficient.

So PR, P and Q is logically equivalent to P. Is logically equivalent to P. If you want, just you can substitute true value and false value. So you will get the result. And what is the second one?

P and here they are different. Symbols are different. So P and PRQ. P and and R.

They are different. it is logically equivalent to 2p it is logically equivalent to 2p so if a problem contains like this we can substitute with p or if the problem contains like this also we can substitute with the p okay now let us see the next one the next one is d mogran's law d morgan's law so this is very very popular law this is very very popular law the first one is negation of prq negation of prq implies Negation P. Negation P. Negation of R means and. Negation of R means and.

And negation Q. Negation Q. B. So if the problem contains negation of P or Q then we can replace with negation P and negation Q. Or if a problem contains negation P and negation Q we can replace with negation P or Q.

According to our problem requirement we have to substitute. RHS in place of LHS or LHS in place of RHS. And the second one is negation of.

This column represents what? Conjunction and whereas this column represents disjunction R. So negation of P and Q. Negation of P and Q is logically equivalent to negation P or negation Q.

Negation P or negation Q. If a problem contains negation of P and Q you can replace with negation P or negation Q. or if a problem contains negation p or negation q you can replace with negation of p and q now let us see the next one the next one is negation law negation or there are several names are there we can call this as negation or complement complement negation or complement law let us see the first one negation if you take p then what is the negation negation p p or negation p if you take p then its negation is negation p and this is nothing but what r loss disjunction so what is the result of p or negation p if you take p as true true or negation p means false true or false means true if you take p as false false or negation false means true so it always returns true as the result and the second law is p and negation p is equivalent to if you take p as true true and negation true true one negation true means false true and false means false if you take p as false false and negation false means true false and true means false we always get false as the result and the last last law is double negation double negation law double negation law negation of negation p is equivalent to negation of negation p is equivalent to p if you take p as true if you take p as true negation true means what false negation of false means true so the true is nothing but p value if you take p as false negation false means true negation of true means false so false is nothing but p value so by using these 10 rules or 10 formulas we can check whether two propositions are equivalent or not without using the truth table so these are the 10 formulas in the next video we'll see various examples regarding this approach.

Transcript for:Understanding Logic Equivalence Laws

Transcript for:
Understanding Logic Equivalence Laws