[Reasoning] Logical Connectives (if, unless, either or) for CSAT, CAT shortcuts formulas approach explained

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Economic Survey
  1. Difference: Syllogism vs Logical connectives
  2. Standard format: logical connectives
  3. Logical connective: if then
  4. Logical connective: Only IF
  5. Logical Connective: UNLESS
  6. Logical connective: otherwise
  7. Logical connective: When, Whenever, every time
  8. Logical Connective: Either OR
  9. Demo Q: Only if: bored TV brother (CSAT 2012)
  10. Demo Q (If, then) Professor Headaches  (CAT’98)
  11. Demo Q: Either or: derailed/late train (CAT’97)

Difference: Syllogism vs Logical connectives

Syllogism (all cats are dog) is a common and routinely appearing topic in most of the aptitude exams (Bank PO, LIC, SSC etc). But Logical connectives is rare. However, in UPSC CSAT 2012 the topic was asked, therefore, you’ve to prepare it.

Syllogism

Logical connectives

Contains words like “all, none, some” etc. Can be classified into UP, UN,PP and PN. Already explained in previous articles. Contains words like “if, unless, only if, whenever” etc. can be classified into 1, ~1, 2, ~2 (we’ll see in this article)
Have to mugup more formulas, takes more time than logical connective questions. Less formulas and quicker than syllogism.
Question Statements:

  1. All cats are dogs
  2. some pigs are cats
  3. no dogs are bird

Conclusion choices:

  1. Some cats are dogs
  2. No birds are cats
  3. some pigs are birds
  4. Some pigs are not birds
Question statements:

  1. I watch TV only if I am bored
  2. I am never bored when I have my brother’s company.
  3. Whenever I go to the theatre I take my brother along.

Conclusion choices:

  1. If I am bored I watch TV
  2. If I am bored, I seek my brother’s company.
  3. If I am not with my brother, than i’ll watch TV.
  4. If I am not bored I do not watch TV.

Standard format: logical connectives

  • If, unless, only if, whenever, every time etc. are examples of Logical connectives.
  • Whenever you’re given a question statement, first rule is: question statement must be in the standard format.
  • The standard format is
  • ****some logical connective word *** simple statement#1, simple statement #2.
  • It means, the question statement must start with a logical connective word, otherwise exchange position. For example
Given question statement Exchange position?
If you’re in the army, you’ve to wear uniform
  • no need because the simple statement containing “IF” is given in the beginning. This is already in the standard format.
You’ve to wear uniform, if you’re in the army
  • We need to exchange position because the part containing “IF” is not given in the beginning of this statement, given statement is not in standard format.
  • Therefore, Rewrite given statement as
  • If you’re in the army, you’ve to wear uniform.
You’ve to salute, whenever Commanding Officer comes in your cabin.
  • Need to exchange position. Because statement doesn’t start with the logical connective “whenever”.
  • Therefore rewrite the given statement as
  • Whenever CO comes in your cabin, you have to salute.

Now let’s derive valid inferences for various logical connectives.

Logical connective: if then

Consider these two simple statements

  1. You’re in army
  2. You’ve to wear uniform.

These are two simple statements. Now I’ll combine these two simple statements (#1 and #2) to form a complex statement.

  • If you’re in army(#1), you have to wear uniform.(#2)

What about its reverse?

  • You’ve wearing uniform (#2)—> that means you’re in the army.(#1)
  • But there is possibility, you’re in navy—-> you’ll still have to wear a uniform. It means,
  • if 1=>2, then 2=>1 is not always a valid inference.
  • Let’s list all such scenarios in a table.
Given statement:If you’re in army(#1), you have to wear uniform.(#2)
Inference? Valid / invalid?
  1. If #2, then #1
If you’ve to wear uniform, you’re in army. you’ve to wear uniform in navy, air force, BSF etc. so this inference is not always valid.
  1. If not #1, then not #2
if you’re not in army, you don’t have to wear uniform. you’ve to wear uniform in navy, air force, BSF etc. so this inference is not always valid.
  1. if not #2, then not #1
If you don’t have to wear uniform, you’re not in army. Always valid.
  • In the exam, you don’t have to think ^that much. Just mugup the following rule:
  • Given statement =“If #1 then #2”, in such situation the only valid inference is “if Not #2, then not #1”.
  • In other words, “if 1st happens then 2nd happens”, in such situation, the only valid inference is “if 2nd did not happen then 1st did not happen”.
  • Now I want to construct a short and sweet reference table for the logical connective problems. So I’ll use the symbol ~= negative.

~1=meaning NOT 1 ( or in other words, negative of #1)

Given Valid inference
If 1, then 2 If not 2, then not 1
If 1=>2 ~2=>~1
  • In some books, material, sites, you’ll find these rules explained as using “P” and “Q” instead of 1 and 2.
  • But in our method, you first make sure the given (complex) statement starts with a logical connective (or you exchange position as explained earlier)
  • We denote the first simple sentence as #1 and second simple sentence as #2.
  • The reason for using 1 and 2= makes things less complicated and easier to mugup.

Logical connective: Only IF

  • In such scenario, you’ve to rephrase given statement into “if then” and then apply the logical connective rule for “if then”.
  • For example: given statement: he scores a century, only if the match is fixed.
  • The “standard format”= only if the match is fixed(1), he scores a century(2).
  • In case of “only if”, we further convert it into an “if” statement, by exchanging positions. That is
  • if he scores a century(#2), the match is fixed(#1).
  • Then apply the formula for “if then” and get valid inference.
  • Here we’ve “if 2=>1” as per our formula for “if then”, the valid inference will be ~1=>~2. Don’t confuse between 1 and 2. Because essentially the valid inference is “negative of end part => negative of starting part”.
  • Therefore “if 2=>1 then ~1=~2”
  • similarly “if 98=>97, then valid inference will be ~97=>~98”
  • Similarly “if p=>q, then valid inference will be ~q=>~p”,
  • similarly “if b=>a, then valid inference will be ~a=~b”) .
  • Update our table
Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=>~1 Negative of end part=> negative of start part
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.

Logical Connective: UNLESS

  • Given statement: Unless you bribe the minister(#1), you will not get the 2G license.(#2)
  • Unless = if…..not.
  • So, I can re-write the given statement as
  • (new) Given statement: If you don’t bribe the minister(#1), you’ll not get the 2G license.(#2)

How to come up with a valid inference here?

#1 You don’t bribe the minister
#2 You’ll not get the 2G license.
  • For “if..then”, We’ve mugged up the rule:  1=>2 then only valid inference is ~2=>~1. (in other words, negative of end part => negative of starting part).
  • let’s construct the valid inference for this 2G minister.
  • we want ~2 => ~1
  • Negative of (2) => negative of (1)
  • Negative of (you’ll not get the 2G license)=>negative of (you don’t bribe the minister)
  • You’ll get the 2G license => you bribe the minister.
  • In other words, If I see a 2G license in your hand, then I can infer that you had definitely bribed the minister.
  • This is one way of doing “unless” questions = via converting it into “if…not” type of statement.
  • The short cut is to mugup another formula: unless1=>2 then ~2=>1.
  • How did we come up with above formula?

Deriving the formula for unless

  • Unless 1=>2 (given statement)
  • if not 1=>2  (because unless=if not)
  • if ~1=>2 (I’m using symbol ~ instead of “not”)
  • ~2=> ~(~1) (because we already mugged up the rule “if 1=>2, then valid inference is ~2=>~1)
  • ~2=>1 (because ~(~1) means double negative and double negative is positive hence ~(~1)=1)

This is our second rule: Unless1=>2 then ~2=>1

Table

Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=>~1 Negative of end part=> negative of start part
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>start part unchanged.

Logical connective: otherwise

  • Suppose given statement is: 1, otherwise 2.
  • you can write it as unless 1 then 2.  (unless1=>2)
  • Then use the formula for “unless.”

Logical connective: When, Whenever, every time

  • Given statement: he scores century, when match is fixed.
  • This is not in standard format of “**logical connective word**, simple statement #1, simple statement #2.”
  • So first I need to exchange the positions: “when match is fixed (#1), he scores century (#2)”.
  • In case of when and whenever, the valid inference is= same like “If, then”. That means negative of end part=>negative of starting part.
  • Same formula works for “whenever” and “Everytime”.
  • Update the table
Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=~1 Negative of end part=> negative of starting part
When When 1=>2
Whenever Whenever 1=>2
Everytime Everytime 1=>2
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>starting part unchanged.

Logical Connective: Either OR

Given statement: Either he is drunk(1) or he is ill(2).

In such cases, if not 1 then 2. And if not 2 then 1.

Meaning,

  1. if he is not drunk then he is definitely ill
  2. if he is not ill, then he is definitely drunk

both are valid. Update the table

Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=~1 Negative of end part=> negative of starting part
When When 1=>2
Whenever Whenever 1=>2
Everytime Everytime 1=>2
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>starting part unchanged.
Otherwise 1 otherwise 2=> rewrite as Unless1=>2.
Either or Either 1 or 2
  • ~2=>1
  • ~1=>2
Negative of any one part=> remaining part remains unchanged.
  • Now let’s solve some questions from old CSAT and CAT papers
  • Please note: in the exam, actual wording / meaning of the simple statement doesn’t matter. Just apply the formulas as given in above table.
  • For example, “if you’re in army, you have to wear uniform.” Then valid inference is ~2=>~1 (you don’t have to wear uniform, then you’re not in army).
  • Now ofcourse there would be exceptional situation when army officer/jawan doesn’t need to wear uniform, for example during espionage mission behind the enemy lines. In that case you don’t have to wear uniform, but you’re still in the army.
  • But keep in mind, while solving logical connective question under the “aptitude/reasoning” portion you don’t have to surgically dissect or nitpick the meaning every statement. Just “if 1=>2” then “~2=>~1”.

Demo Q: Only if: bored TV brother (CSAT 2012)

Examine the following statements:

  1. I watch TV only if I am bored
  2. I am never bored when I have my brother’s company.
  3. Whenever I go to the theatre I take my brother along.

Which one of the following conclusions is valid in the context of the above statements?

  1. If I am bored I watch TV
  2. If I am bored, I seek my brother’s company.
  3. If I am not with my brother, then I’ll watch TV.
  4. If I am not bored I do not watch TV.

Approach

First we’ll construct valid inferences from the question statements

Given Question Statement #1:

  • Given =I watch TV only if I am bored
  • This is not in standard format. So first exchange position
  • Only if I’m bored (1), I watch TV(2)
  • What is the valid inference? Just look at the formula table
  • Only if 1=>2 then ~1=~2
  • Valid inference= if I’m not bored, I do not watch TV.
  • Look at the statements given in the answer choices, (D) matches. Therefore, final answer is (D).

Demo Q (If, then) Professor Headaches  (CAT’98)

You’re given a statement, followed by four statements labeled A to D. Choose the ordered pair of statements where the first statement implies the second and two statements are logically consistent with the main statement.

Given statement: If I talk to my professors(1), then I didn’t need to take a pill for headache.(2)

Four Statements

  1. I talked to my professors
  2. I did not need to take a pill for headache
  3. I needed to take a pill for headache
  4. I did not talk to my professor.

Answer choices

  1. AB
  2. DC
  3. CD
  4. AB and CD

Approach

Given statement is in standard format already

#1 I talk to my professors
#2 I didn’t need to take a pill for headache.

Let’s classify the four statements

Classification Four statements
1
  1. I talked to my professors
2
  1. I did not need to take a pill for headache
~2
  1. I needed to take a pill for headache
~1
  1. I did not talk to my professor.

Answer choice (i) AB

If you observe the answer choice (I): AB= I talked to my professors, I did not need to take a pill for headache. This is valid because if 1=>2 is already given in the question statement itself.

Answer choice (ii) DC

  • I did not talk to my professor (~1), I needed to take a pill for headache (~2). Meaning ~1=>~2.
  • This is invalid because as per our table, if 1=>2, then valid inference is ~2=>~1.

Answer choice (iii) CD

I needed to take pill for headache (~2), I did not talk to my professor (~1). Meaning ~2=>~1. This is valid as per our table. Therefore final answer is (IV) AB and CD

Demo Q: Either or: derailed/late train (CAT’97)

Given statement: either the train is late (1) or it has derailed (2)

Four statements

  1. Train is late = 1
  2. Train is not late = ~1
  3. Train is derailed =2
  4. Train is not derailed =~2

(^note: I’ve classified the statements in advance)

Answer choice

  1. AB
  2. DB
  3. CA
  4. BC

Approach

As per our table, the valid inferences for either or are

~2=>1 If the train is not derailed, it is late. DA
~1=>2 If the train is not late, it is derailed BC

Correct answer is (III): BC
For more articles on reasoning and aptitude, visit Mrunal.org/aptitude

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197 Comments on “[Reasoning] Logical Connectives (if, unless, either or) for CSAT, CAT shortcuts formulas approach explained”

  1. Another question

    when i see horror movie i had a bad dream
    a) i saw a horror movie
    b) i didnt saw a horroe movie
    c) i didnt had a bad dream
    d) i had a bad dream

    option :
    1) CB
    2)AD
    3) BC
    4) AC

    Answer According to TMH – a, but m confused b/w option 1 & 2

    1. ansr CB ~2=>~1 ,,v hv to mk inference. 2nd option is just qusn statemnt but if in ansr they gv both ,,,tick both oderws CB.

    2. restatement is never a valid answer, its neither a conclusion nor a reason.

  2. @anurag……. option 3 in sleep wala ques.
    N both 1 n 2 are correct in bad dream ques.

  3. Hello Sir thanks a lot for simplifying sylloligism. Sir, Somewhere in ur blog u mentioned abt tool which remove unwanted texts before taking printout, so that it will save lot of pages. could you please mention abt tht tool so tht we can save no of pages and also money :-) while taking printout.

  4. @parthi

    Basically if we make the venn diagram of both the options they would be same. I guess this seems like a bug with with venn diagram method. Well, but wat seems more appropraite is if rivers flows c cranks and not the other way round as see…
    If A then B cannot be stated as only if B then A when i guess their Venn diagrams would be identical. Let me know!

  5. Thanks a lot….

  6. Wat do we infer from ” all except a few” some are and simultaneously some are not?

  7. Please help…
    1)some who study will not become graduates.
    2)To become graduates student must study.
    3) Only students can become graduates.

    conclusion:-
    a)Some who become graduates are not student.
    b)All students who study become graduates.
    c)Some who study and become graduates are students.
    d)Students who do not study will not graduate.

    1. C & D can be concluded becuase the statement 2 says To become Graduates students MUST study. and 3rd says that only Students can become Graduates.
      In Mathematics-
      That means Graduate is a subset of Study and Student both and
      Study is a subset of Student.
      so there are 3 concentric circles with Student the outer most, then Study and the inneromst circle is Graduate.

      In simple words (i m trying)-
      A) Graduates are students first,grads second Why ? because state 3 says ONLY studs can become Grads. Hence this cant be concluded.
      B) All students who study cant be sure that they will DEF Graduate (just as 17lakh applicants gave SBI but only 1500 will be selected), anyways, because Statement1 says Some who study still not Graduate. Hence this cant be concluded.
      C) Since the 3rd statement says , that whosover is Grad he is a Student too and State 2 says that Stud must have studied to become Grad. Hence this can be concluded.
      D) Students who dont study = Can never Graduate. Why ? Because st2 says They MUST study but its not necessary that they will DEFINITELY graduate. but they stand a chance. Hence this can be concluded.

      I hope it helps. Do tell the Answer too.

      1. Except “A” other conclusions r correct. 3 concentric circles ll not come. Please check. B can be concluded as second statement says “2)To become graduates student must study.”. so students if they study ll become graduate. (SBI s a competitive xam so all will not clear, but any student can become graduate provided he studies. NOTE :Even i am a graduate =D )..

        1. Vikky

    2. i dont think C can be concluded because

      Only Students = Graduates (and no other)
      quantifier in C is “some” it has to be “ALL”
      Lets say i am doctor and i study
      if i become graduate i must have been Student. so not some but All

      ans is D

      ANSWER??

      1. Your point is correct but here its not that Statement cant be concluded.

        Whenever its given –

        “All Roads are infrastructure.”

        Then you can always conclude that – “Some Roads are infrastructure.”
        Its not necessary that if ALL is given then the conclusion must have all. It can obviously have “SOME” .

        1. i disagree,

          All roads are infra. subject is distributed.. its not logically correct to blur the space boundary
          though true in common sense

          some roads are infra cant be “concluded” because in this statement neither subject nor object is distributed..

          as roads is Subset of infra we can say that some infra is roads. it not a conclusion too because its an INVERSE

          BHAI JISNE POOCHA HAI QUESTION WOH ANSWER TO BTA DOOOO…….

          1. I think it is C and D both. And the explaination of manu jha is correct. pls correct answer?

  8. sir, in question..you can see the star only if you go to cinema…correct answer should be i didnt see the star i didn’t go to cinema..bt reverse is givn…please help me with this thnku…

    1. i got the solution..thnku..

  9. You are the king,, mrunal…

  10. mrunal, can i infer from your explanation of “either – or” type-
    that answer format will be :-
    -1(negative of first part)and 2(unchanged second part)
    ?????

  11. Sir, it is very interesting and very good elaboration. i read a book of CAT (author name started by a. not mentioning full name). but there i does not find the same thing as u wrote . thanks ………..

  12. thanx for the explanation…it will be nice if u elaborate on more interconnected sentence for
    e.g If ankita eats pastry, then it is a black forest or a pineapple.

    plz xplain all the possible outcome.

  13. hello Admin,
    I am new to this site. For the info above discussed there are other valid outcomes as well.
    Only if 1=> 2 it also mean 2=>1 apart from ~1=>~2. Please confirm it.

  14. Dear Mrunal Sir,
    Your methods to derive conclusions only in negatives like not 1 not 2 types .
    I faced a lot of problem while facing questions based on non negative conclusion.One more doubt Can I derive another conclusion based on two already derive conclusions.
    Q.1
    1.He writes whenever he is angry.
    2.It is cloudy only if he is angry.
    3.He is angry only if he is hungry.
    Conclusion
    A.He writes whenever he is hungry.
    B.He is hungry if he writes.
    C.He writes only if it is cloudy.
    D.Whenever it is cloudy,he writes.
    Q-2
    1.There are as many engineers in this organisation as there are doctors.
    2.Only if a doctor agrees to a plan will an engineer agree to plan.
    3.An engineer agrres to a plan only if the majority of engineers including him agree to it.
    Conclusions
    A.Only if an engineer agree to a plan will a doctor agree to it.
    B.An engineer will agree to plan if all doctor agree to it.
    C.An engineer might agree to a plan if the total number of engineers agreeing to a plan is more than half the total numbers of doctors.
    D.If all doctors agree to a plan,all engineers agree to it as well.

    Reply asap.

  15. Hello Sir
    although option D)If I am not bored I do not watch TV. (csat 2012)
    can definitely be concluded.
    why not option
    B)If I am bored, I seek my brother’s company.
    bcoz I am bored = negative of ( i am not bored) also
    I SEEK my brother company = does not SEEK here mean that HE IS NOT
    IN HIS BROTHER’S COMPANY.
    IF AM WRONG PLEASE CORRECT….

  16. i really appreciate ur contribution for us sir .every time i read ur article i just say -u r really too good and in my view best tr .

  17. Hw can i Solve ths:::

    You cannot clear the CSAT unless you are intelligent
    1 u r intelligent
    2u can clear the CSAT
    3 u r nt intelligent
    4 u cannot clear the CSAT

    answer is :: C……….. How???? Plz help me!!

    1. tell us all the options first.
      how do we know what is A , B, C and D?

      1. thnx for reply…
        You cannot clear the CSAT unless you are intelligent
        a u r intelligent
        b u can clear the CSAT
        c u r nt intelligent
        d u cannot clear the CSAT

        conclusion: are bd, ac, cd, ab, ; given ans is cd, how? accordingly above, valid interference is ~2=>1, ~2 cancels negative sentence ; is m right????

        plz

        1. look…it means if u cleared the csat, u r intelligent….ie all csat are intelligent….draw a venn with intelligent encircling csat and u get ur answer…not intelligent means no csat,,,,

        2. we can write it as
          you can clear CSAT only if you are intelligent
          means if you cleared CSAT then definitely you are intelligent but if you are intelligent you cannot say that you can definitely clear CSAT
          also if you are not intelligent then you cannot clear CSAT but it does not mean that if you have not cleared CSAT then you are not intelligent i.e. (you may be intelligent)

          1. so answer is CD

    2. luk aditi
      You cannot clear the CSAT unless you are intelligent
      U CAN WRITE DYS AS
      unless you are intelligent,You cannot clear the CSAT
      NOW ABOVE SENTENCE CAN BE WRITTEN IN D FRM LYK DAT
      UNLESS => IF…NOT
      so …….if you r nt intellignt ,u cannot clear CSAT
      NOW SIMPLY APPLY D LOGIC OF IF NOT
      2 SOLUTIONS R POSSIBLE
      RULE A…~2=~1
      RULE B…1=2
      SO WE WILL;TOOK CD as a answer by applyn RULE B
      on d other syd other options not CRRCT
      HOPE U UNDRSTND D SOLUTION

  18. Dear sir,
    Whether in the last example ,AD and DA are both possible. Please enlighten us :)

  19. Sir,
    How to solve a question which do not contain any “if” and “only if”. Below is the question asked in CL test series:

    Statement: There are as many engineer in the organization as there are doctors.

  20. another question which i am unable to solve is following:

    statements are:
    1. whenever prices goes up , farmers are affected
    2. Farmers are affected only if it does not rain
    3. It rains if there are clouds
    Which of the following conclusion can be drawn:
    a) Farmers are always affected by rains,
    b) If it does not rain Farmers are affected
    c) Whenever there are clouds prices go up
    d) If there are clouds farmers are not affected

    Correct Answer is D, But how and how to solve option A?

  21. 1) Prices up —-> Farmers affected (FA)
    2) NO rain —–> FA
    3) Clouds —-> Rains

    So,
    OP 1) Does not qualify as —- FA by Price rise also
    OP 2) Does not qualify as —- FA by Price rise also
    OP 3) No relation between the three
    OP 4) Correct – coz CLOUDS —-> Rains —-> Farmers NOT affected.

    Hope this helps.

    thanks

    1. Thanks for your reply but kindly explains below related queries:
      1. as per universal rule.. only if p then q : q->p && ~p->~q so deduction of statement 2 [“Farmers are affected only if it does not rain”] would be as follows
      FA -> No rain && Rain->no FA

      but you have derived it as ” 2) NO rain —–> FA “… HOW??

  22. pretty easy mate…… u can solve simply by logic… statement 2 is antithema of conclusion 1,,,
    more systematic approach…1. price imples farmers affection ie all prices are farmers… 2 //farmers implies no rain…3 cloud imples rain that all cloud are rain…
    now draw venn with prices encircled by farmers in turn encircled by no rain and separate circle of cloud u get ur answer

  23. ask any csat question… can ask gs as well

  24. question -if P runs for president, C does not.
    answers 1.P runs for president,C does not.
    2.C runs for president,P does not.
    3.both P and C run for president.
    4.neither P nor C run for president.
    which of the above 4 statements are correct.

  25. question -if P runs for president, C does not.
    answers 1.P runs for president,C does not.
    2.C runs for president,P does not.
    3.both P and C run for president.
    4.neither P nor C run for president.
    which of the above 4 statements are correct.

  26. SUPERB EXPLAINATION……. YET NEED OF SOME MORE EXAMPLES TO SOLVE N PRACTICE SR

  27. for Unless, is it ~ 2=> 1 or ~2=>~1 , please clarify

  28. sorry got it, thanks

  29. hi sir want to prepare for the post of probationary officer and rbi grade b officer so please suggest me the best practise paper and also economics and banking terms…its my humble request to u and thanking you

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