First time here? Checkout the FAQ!
+3 votes

Choose the correct alternatives (More than one may be correct).

Indicate which of the following well-formed formulae are valid:

  1. $\left(P\Rightarrow Q\right) {\wedge} \left(Q \Rightarrow R\right) \Rightarrow \left(P \Rightarrow R\right)$
  2. $\left(P\Rightarrow Q\right) \Rightarrow \left( \neg P \Rightarrow \neg Q\right)$
  3. $\left(P{\wedge} \left(\neg P \vee  \neg Q\right)\right) \Rightarrow Q$
  4. $\left(P \Rightarrow R\right) \vee \left(Q \Rightarrow R\right) \Rightarrow \left(\left(P \vee Q \right)  \Rightarrow R\right)$
asked in Mathematical Logic by Veteran (30.3k points)  
retagged by | 217 views

A is true it is Hypothetical syllogism.


In classical logic, hypothetical syllogism is a valid argument form which is a syllogism having a conditional statement for one or both of its premises. If I do not wake up, then I cannot go to work. If I cannot go to work, then I will not get paid. Therefore, if I do not wake up, then I will not get paid.



[(P⇒Q)∧(Q⇒R)]      =====>      (P⇒R)

----------------------                        -----------

      T                                         F

You Can Not find   T====>F

So this formula is always true

3 Answers

+5 votes
To prove any wff valid or tautology try to use this analogy.

Since implication A->B is FALSE only when A=T and B=F.So to prove any implication is valid or not try to get TRUE->FALSE if we succeed then it is not valid,if we not then wff is valid.

So for option A

substitute P=T and R=F

RHS P->R become FALSE

LHS (P->Q)^(P->R)

To get true here we need T^T so substitute Q=T which makes P->Q TRUE and P->R FALSE so T^F=F which makes LHS=FALSE.

Hence we are unable to get T->F which proves wff given in OPTION A is valid.

NOTE: we can use similar kind of logic to prove contradiction.
answered by Active (2.1k points)  
+1 vote
Create a logic table for each of the given formula and check if we are getting true everywhere.
answered by Veteran (15.2k points)  
0 votes

Ans) A

For reason follow pics

answer part1

answered by (77 points)  

Top Users Jul 2017
  1. Bikram

    3782 Points

  2. manu00x

    2464 Points

  3. Debashish Deka

    1832 Points

  4. joshi_nitish

    1494 Points

  5. Arnab Bhadra

    1096 Points

  6. Arjun

    1054 Points

  7. Hemant Parihar

    1050 Points

  8. Shubhanshu

    972 Points

  9. Ahwan

    876 Points

  10. akash.dinkar12

    642 Points

23,953 questions
30,895 answers
29,272 users