Inequalities, "Only a Few" Syllogisms and Logical Inference
Coded Inequalities and Either-Or
In a coded inequality the relations are replaced by symbols, and the stem defines them, because each paper uses its own code. The code used in this chapter is: $ means ≥, % means ≤, # means >, @ means <, and & means =. Decode first, then read the chain. Do not try to reason with the symbols themselves; the extra step costs five seconds and removes the commonest error, a sign mix-up.
Worked example. Statements: R # S, S $ T, T & U. Decoded: R > S ≥ T = U. Conclusion P: R > U follows, since R > S is strict and the rest is a same-direction path. Conclusion Q: S > U does not follow, because S ≥ T = U gives only S ≥ U. The answer is "Only P follows".
Either-or. Two conclusions about the same pair give "either P or Q follows" when neither follows alone but one of them is certain to hold, whatever the values are. In this chapter that is used only for complementary pairs: X > Y with X ≤ Y, and X ≥ Y with X < Y. Such a pair covers every case between X and Y, so exactly one is true. The pair X > Y and X < Y is not complementary, because X = Y is left out.
(1) neither conclusion follows alone;
(2) both are about the same two letters;
(3) the pair is complementary. If the two letters are not linked by any path at all, all three relations are possible, so a complementary pair is the right answer. If they are linked by a path, one conclusion usually follows and the question is not either-or.
Syllogisms with "Only a Few" and "At Least Some"
A syllogism item gives statements about groups and asks which conclusions definitely follow. The bank forms are "all", "no", "some" (also written "at least some"), "some … not" and "only a few". This chapter fixes the convention so that no item is ambiguous. Only a few X are Y means both that some X are Y and that some X are not Y. A named group is taken to be non-empty, so "All X are Y" also gives "some X are Y".
| Statement | What definitely follows | What does not |
|---|---|---|
| Some A are B | Some B are A | Some A are not B |
| Only a few A are B | Some A are B; some A are not B; some B are A | Some B are not A |
| All A are B | Some A are B; some B are A | All B are A |
| No A is B | No B is A; some A are not B | Some A are B |
Two statements together. Join them through the middle term, using the rules below, all checked by listing every Venn arrangement that fits the statements. A conclusion that holds in all arrangements follows; one that fails in even a single arrangement does not.
| Statements | Definite conclusion | Not definite |
|---|---|---|
| Some A are B; all B are C | Some A are C | All A are C |
| Only a few A are B; all B are C | Some A are C; some A are not B | Some C are not A |
| Only a few A are B; no B is C | Some A are not C | No A is C |
| All A are B; no B is C | No A is C | Some C are A |
| Only a few A are B; some B are C | Nothing about A and C | Some A are C |
Worked example. Only a few flats are houses. All houses are buildings. Some flats are buildings follows, since some flats are houses and every house is a building. Some flats are not houses follows from "only a few flats are houses". Some buildings are flats follows as the reverse. Some buildings are not flats does not follow, because every building might be a flat or a house.
Reading Conclusions: Definite, Either-Or and NOT-Follow
Test a conclusion by trying to break it. Draw the arrangement that fits the statements and makes the conclusion as unlikely as you can. If you can still draw one valid picture in which the conclusion is false, it does not follow. A "some" conclusion needs a guaranteed shared member: it comes from a chain of "all" statements, or from a "some" (or "only a few") statement continued only by "all" statements. Two "some" statements, or "all A are B" followed by "some B are C", never guarantee a shared member between the outer groups.
Either-or in syllogisms. If two groups have no stated link and the conclusions are "some X are Y" and "no X is Y", neither follows alone, but in every arrangement one of them holds, so "either P or Q follows". Example: all pens are books, and some books are cups. Is any pen a cup? It may or may not be; both "some pens are cups" and "no pen is a cup" are possible, and exactly one is true.
Logical Inference from Statements and Conditions
A statement-inference item asks what is certain given a statement. Allow only what the statement says or strictly entails, and reject what is merely likely, is the converse, or brings in outside facts. For quantities, use the arithmetic of overlaps: if 60 per cent of customers use mobile banking, 70 per cent use net banking and every customer uses at least one, then the share using both is 60 + 70 − 100 = 30 per cent.
| Rule | Valid inference | Invalid (often offered) |
|---|---|---|
| If p then q | If not q then not p | If q then p; if not p then not q |
| p only if q | If p then q (p needs q) | If q then p (q is enough for p) |
| p if q | If q then p | If p then q |
| p unless q | If not q then p | If q then not p |
Worked example. "A loan is sanctioned only if the applicant's score is at least 700." This is "if sanctioned then score at least 700". If Deepa's loan was sanctioned, her score is at least 700. If Sunil's score is 750, nothing follows: the bank needs the score but may also need other conditions. "At least 700" is not "above 700", and a loan may well be sanctioned at exactly 700.
Decision-Making Conditions
A decision-making item gives a rule sheet and a case, and asks what happens to the case: selected, rejected or referred. List the conditions as a checklist and test them in the order given, since a later rule often applies only to cases that pass or fail an earlier one. The commonest loss is the boundary word. "At least" and "up to" include the number; "more than" and "below" exclude it; "between 21 and 30" is inclusive only when the stem says so.
| Phrase | Includes the number? | For 700 |
|---|---|---|
| at least 700; 700 or more | Yes | 700 qualifies |
| more than 700; above 700 | No | 700 fails |
| up to 700; at most 700 | Yes | 700 qualifies |
| below 700; less than 700 | No | 700 fails |
Worked example. Rule: a candidate is shortlisted if he or she is a graduate with at least 55 per cent and is aged 21 to 30 years (both included). A graduate with at least 55 per cent aged 31 or 32 is referred to the committee. All others are rejected. Case 1: graduate, 54 per cent, age 25. The marks condition fails, so the case is rejected. Case 2: graduate, 58 per cent, age 32. Age is outside 21 to 30, but the referral rule covers 31 and 32, so the case is referred. Test every rule against the case; do not stop at the first that fails.
Key takeaways
- In a chain, a relation between the ends follows only if all links point the same way, and it is strict only if at least one link is strict; arrows that meet at a middle letter give no relation.
- Decode coded inequalities first using the code in the stem; either-or applies to a complementary pair (X > Y with X ≤ Y, or X ≥ Y with X < Y) when neither follows alone.
- "Only a few A are B" means some A are B and some A are not B; it gives "some B are A" and "some A are not B" but never "some B are not A".
- A "some" conclusion needs a guaranteed shared member: two "some" statements, or "all A are B" then "some B are C", give nothing about A and C.
- "p only if q" means if p then q; the contrapositive is valid, the converse and inverse are not; "at least" includes the number and "more than" does not.
- For decisions, test every rule in the order given and watch the boundary words; the sheet's last rule decides cases that nothing else covers.
Practice questions (16)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
In the statement M ≥ N > O = P, which conclusion definitely follows?
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Answer: C — M > P
The chain M ≥ N > O = P runs downward with N > O strict, so M > P. M = P is not possible because of the strict link, and P ≥ N reverses the direction. O > M also reverses it.The symbols are defined as: # means >, @ means <, and & means =. If W # X & Y @ Z, which relation between W and Z is definitely true?
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Answer: D — No definite relation
Decoding gives W > X = Y < Z. The arrows meet at Y, so W and Z are both above Y but have no order between them: all of >, < and = are possible. W > Z is the tempting guess because the chain starts with a >.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statement: Only a few doctors are teachers. Which of the following definitely follows?
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Answer: A — Some doctors are not teachers
By the convention, only a few doctors are teachers means some doctors are teachers and some doctors are not. The claim about teachers not being doctors is a reverse overreach: all teachers may well be doctors. The claims that all teachers are doctors or that no teacher is a doctor are not guaranteed, and the latter contradicts the statement.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: Only a few pencils are erasers. All pencils are sharpeners. Conclusions. P: Some erasers are sharpeners. Q: Some sharpeners are not erasers. Which conclusions follow?
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Answer: B — Both P and Q follow
Some pencils are erasers, and those pencils are sharpeners, so some erasers are sharpeners (P). Some pencils are not erasers, and these are sharpeners, so some sharpeners are not erasers (Q). Both are guaranteed by the two statements, so each of the other options drops a conclusion that does follow.Rule: A bank sanctions a loan only if the applicant's credit score is at least 700. Deepa's loan was sanctioned. Which of the following is certain?
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Answer: C — Deepa's credit score is at least 700
"Only if" says the score is needed: sanctioned implies a score of at least 700. Exactly 700 and above 700 are each only possibilities, not certainties. The claim that every applicant with at least 700 gets a loan reverses the rule, treating the score as sufficient when it is only necessary.A loan application is approved if the monthly income is at least ₹40,000 and the credit score is at least 700. If the income is at least ₹40,000 but the credit score is from 650 to 699 (both included), the application is referred to the branch manager. In every other case it is rejected. Neha's monthly income is exactly ₹40,000 and her credit score is 699. What happens to her application?
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Answer: A — It is referred to the branch manager
Income of exactly ₹40,000 satisfies "at least ₹40,000", and 699 is inside the included range 650 to 699, so the referral rule applies. Approval needs a score of 700. Rejection because the income is not above ₹40,000 misreads "at least" as "above", and a plain rejection ignores the referral rule.The symbols are defined as: $ means ≥, % means ≤, # means > and & means =. Statements: B $ D, D # F, F & H, H % J. Conclusions. P: B > H. Q: J > F. Which conclusions follow?
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Answer: D — Only P follows
Decoded: B ≥ D > F = H ≤ J. B ≥ D > F = H gives B > H, so P follows. J ≥ H = F gives only J ≥ F, with no strict link, so Q does not follow. Q is the usual trap, since the chain contains a strict sign elsewhere but not on this path.The symbols are defined as: $ means ≥, % means ≤, & means =, # means > and @ means <. Statements: M $ N, N & O, O % K. Conclusions. P: M # K. Q: M % K. Which of the following is correct?
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Answer: A — Either P or Q follows
Decoded: M ≥ N = O ≤ K. M and K are both above or equal to the middle value but have no path between them, so M > K, M = K and M < K are all possible. P (M > K) and Q (M ≤ K) are complementary, so neither follows alone yet one must be true.Statements: D ≥ E, F ≥ E, F > G. Conclusions. P: D ≥ G. Q: G < E. Which conclusions follow?
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Answer: B — Neither P nor Q follows
D and F are both above or equal to E, and G is below F, but there is no path from D to G or from G to E. D ≥ G can be false when G is large, and G < E can be false when G is greater than E. Neither follows.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: Some funds are bonds. Only a few bonds are shares. All shares are deposits. Conclusions. P: Some bonds are deposits. Q: Some funds are shares. Which conclusions follow?
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Answer: C — Only P follows
Some bonds are shares and all shares are deposits, so some bonds are deposits (P). The funds that are bonds need not be the bonds that are shares, so Q is not guaranteed: it chains two "some" links through bonds.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: Only a few shares are bonds. Some bonds are funds. No fund is a deposit. Conclusions. P: Some shares are funds. Q: Some bonds are not deposits. Which conclusions follow?
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Answer: D — Only Q follows
Some bonds are funds and no fund is a deposit, so those bonds are not deposits (Q). P joins two "some" links (shares-bonds, bonds-funds), so the shares that are bonds need not be the bonds that are funds, and P is not guaranteed.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: Only a few tellers are cashiers. Only a few cashiers are managers. Conclusions. P: Some tellers are managers. Q: Some managers are not cashiers. Which conclusions follow?
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Answer: A — Neither P nor Q follows
The statements link tellers and managers only through cashiers, with two "only a few" links, so P is not guaranteed. Only a few cashiers are managers says some cashiers are not managers, not that some managers are not cashiers, so Q fails too: every manager might be a cashier.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: All cheques are instruments. Some instruments are drafts. Conclusions. P: Some cheques are drafts. Q: No cheque is a draft. Which of the following is correct?
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Answer: B — Either P or Q follows
The drafts among the instruments may or may not include cheques, so neither P nor Q is certain alone. But "some cheques are drafts" and "no cheque is a draft" are complementary: one of them must hold. Neither is therefore the wrong choice for an unlinked pair like this.Convention: "only a few X are Y" means some X are Y and some X are not Y; every group named is taken to have at least one member. Statements: Only a few lamps are fans. All fans are coolers. No cooler is a heater. Which of the following does NOT follow?
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Answer: A — Some fans are not lamps
Some lamps are fans, and fans are coolers, so some lamps are coolers; those lamps are fans and so not heaters; and no fan is a heater. But only a few lamps are fans does not say that some fans are not lamps, since all fans may be lamps. That is the statement that does not follow.In a survey of a branch's customers, 60 per cent use mobile banking and 70 per cent use net banking. Every customer uses at least one of the two. What per cent of the customers use both?
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Answer: B — 30
Everyone uses at least one, so 60 + 70 − both = 100, which gives both = 30. Ten is the difference of the two shares, and 40 is the share that uses only net banking plus the share that uses both, which does not answer the question.Rule: If the server is down, no transaction is processed. Some transactions were processed on Monday. Which of the following must be true?
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Answer: D — The server was not down on Monday
This is the contrapositive: some transaction was processed, so it is false that no transaction was processed, so the server was not down. The claim that the server is down whenever no transaction is processed is the converse, which is not guaranteed because transactions can stop for other reasons.