47 votes 47 votes Let $a, b, c, d$ be propositions. Assume that the equivalence $a ⇔ ( b \vee \neg b)$ and $b ⇔c$ hold. Then the truth-value of the formula $(a ∧ b) → (a ∧ c) ∨ d$ is always True False Same as the truth-value of $b$ Same as the truth-value of $d$ Mathematical Logic gatecse-2000 mathematical-logic normal propositional-logic + – Kathleen asked Sep 14, 2014 edited Jun 20, 2017 by Silpa Kathleen 12.0k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply strawberry-jam commented Nov 7, 2022 reply Follow Share $a \leftrightarrow (b \lor \neg b) = a \leftrightarrow T \Rightarrow a = T$ $\therefore (a \land b) \rightarrow (a \land c) \lor d = b \rightarrow c \lor d = \neg b \lor b \lor d = T$ Ans: (A.) 1 votes 1 votes Shukla_ commented Mar 31, 2023 reply Follow Share . 0 votes 0 votes Please log in or register to add a comment.
Best answer 55 votes 55 votes Given that $\ a\Leftrightarrow (b\vee \neg b)$ and $ b\Leftrightarrow c$ Now, $(a\wedge b)\rightarrow (a \wedge c)\vee d$ $\equiv (a\wedge b)\rightarrow (a \wedge b)\vee d\;\;\;\; (\because b \Leftrightarrow c)$ $\equiv \neg (a\wedge b)\vee (a \wedge b)\vee d$ $\equiv T \vee d$ $\equiv T$ Hence, Option(A) True. LeenSharma answered May 16, 2017 edited Jan 13, 2023 by shadymademe LeenSharma comment Share Follow See all 5 Comments See all 5 5 Comments reply sid1221 commented Jun 11, 2017 reply Follow Share @ leen how did you change c to b? 1 votes 1 votes VishalBarkule commented Jun 11, 2017 reply Follow Share @sid1221 it is given in question that 'b' is equivalent to 'c'(b$\Leftrightarrow$c) 0 votes 0 votes sid1221 commented Jun 11, 2017 reply Follow Share o yes thanks ..i missunderstood 0 votes 0 votes Ayushak commented Jun 10, 2021 reply Follow Share It is given that a⇔(b∨∼b) which implies a⇔T so there’s nothing wrong if we put a as T right? 0 votes 0 votes Sudhanshu10 commented Jun 24, 2021 reply Follow Share Yes, we can use a=T and b=c. 0 votes 0 votes Please log in or register to add a comment.
49 votes 49 votes Given that, ab∨~b It is equivalent to aTRUE (a∧b)((a∧c)∨d) wkt, 1∧x = x (a∧b) = 1∧b = b similarly, 1∧c = c We now have, b (c∨d) Which can be written as, ~b∨c∨d We also know that bc ~b∨c = TRUE TRUE∨d = TRUE And hence answer is option a Gate_15_isHere answered Jan 25, 2015 Gate_15_isHere comment Share Follow See all 3 Comments See all 3 3 Comments reply sameer2009 commented Jul 17, 2015 reply Follow Share Explanation is correct, just to highlight the following statement We also know that bc holds ~b∨c = TRUE if you mean bc means ~b∨c = TRUE then its not true. Although you may not mean this, but while reading the answer it seems that. 6 votes 6 votes Lakshman Bhaiya commented May 2, 2017 reply Follow Share Nice Explanations 0 votes 0 votes Anurag Tiwari 1 commented Oct 5, 2019 reply Follow Share great explanation 0 votes 0 votes Please log in or register to add a comment.
16 votes 16 votes a ⇔ ( b V ~b) = and a ⇔ True means both a and True are equivalent b ⇔c means both b and c are equivalent (a ∧ b) → (a ∧ c) ∨ d =(True ∧ b) → (True ∧ c) ∨ d (a ⇔ True) = b → c ∨ d = ~b ∨ c ∨ d = ~b ∨ b ∨ d (b ⇔c) = True ∨ d =True Hence ans is A Gate Ranker18 answered Jun 19, 2017 edited Jan 4, 2018 by Puja Mishra Gate Ranker18 comment Share Follow See all 0 reply Please log in or register to add a comment.
7 votes 7 votes This will be helpful. see my solution: Sonu Kumar 1 answered Nov 16, 2017 edited Nov 16, 2017 by Sonu Kumar 1 Sonu Kumar 1 comment Share Follow See all 2 Comments See all 2 2 Comments reply Manis commented Nov 30, 2017 reply Follow Share Nyc..... 0 votes 0 votes Puja Mishra commented Jan 4, 2018 reply Follow Share Nice hand writing ... 0 votes 0 votes Please log in or register to add a comment.