EPFO EO/AO 2023 General Ability Test · Question 49 of 120

Correct — C, (c) A – b = b – A. The booklet prints the 'not' of the stem in bold italic, so the item asks for the statement that is NOT a law, and subtraction is the operation that fails. An operation is commutative when swapping the two operands leaves the result unchanged. Addition and multiplication have that property; subtraction does not, and a single counterexample settles it: five minus three is two, while three minus five is minus two. The two sides are equal only in the special case where the operands are themselves equal — set A minus b equal to b minus A and rearranging gives twice A equal to twice b, so A must equal b. An identity has to hold for every value of its symbols, and a statement that holds only when the two symbols happen to coincide is an equation to be solved, not a law. The same conclusion follows from the other direction the paper may intend. In Boolean algebra, which is where a computers paper usually places a question of this shape, the commutative laws are stated for exactly two operations — the sum, A plus B equals B plus A, and the product, A times B equals B times A. There is no subtraction in Boolean algebra at all; the operations are complement, sum and product. So the statement in option (c) is not a commutative law under that reading either, this time because the operation it names does not exist in the system. One feature of the printed options is worth naming, because it looks like a trap and is not. The booklet mixes upper and lower case — option (a) is written with a lower-case second symbol and option (d) with an upper-case one — so those two options state the same commutative law of multiplication in two different typographies. Both are true, so neither can answer a question asking which statement fails, and the case of the letters changes nothing about the mathematics.

• (a)A * b = b * A — This IS a commutative law, so it cannot answer a 'not' stem. It states commutativity of multiplication — or, read in Boolean algebra, of the logical product, the AND operation, whose truth table is symmetric in its two inputs. The lower-case second symbol is a typographical feature of the booklet and has no mathematical significance: a name is a name, and the law holds for whatever the two symbols stand for.
• (b)A + b = b + A — This IS a commutative law — of addition in ordinary arithmetic, and of the logical sum, the OR operation, in Boolean algebra. It is the statement candidates are surest of, which makes it a safe option to eliminate first and then a dangerous one to linger over. Note that commutativity of addition does not by itself carry any of the other laws with it: an operation can be commutative without being associative, and the two must be checked separately.
• (d)A * B = B * A — This IS a commutative law, and it is the same law as option (a) with the second symbol printed in upper case. Both are true statements, which is the point: a question asking which statement is NOT a law can afford to offer the same true statement twice, because the answer is decided by the one option that fails. A candidate who spots the near-duplication should read it as a signal that neither of the pair is the answer.

Three structural laws are asked about together and are worth separating cleanly. Commutativity concerns the ORDER of two operands: the operation gives the same result either way round. Associativity concerns the GROUPING of three: it does not matter which pair is combined first. Distributivity links two different operations, allowing one to be spread across the other. Addition and multiplication of numbers are commutative and associative; subtraction and division are neither. Boolean algebra, the system underlying digital logic, defines three operations — complement, logical sum and logical product — and both of its binary operations are commutative and associative, with each distributing over the other, which is a symmetry ordinary arithmetic does not have. Its further laws are the ones examiners return to: the identity laws, the complement laws, the idempotent laws saying a quantity summed or multiplied with itself gives itself back, absorption, and De Morgan's theorems, which convert the complement of a sum into the product of complements and the complement of a product into the sum of complements. Familiar non-commutative operations outside arithmetic are worth carrying as well — matrix multiplication, composition of functions, division, exponentiation and set difference all give different answers when the operands are swapped.

This item sits in the computers block rather than the mathematics block, which is a hint about the intended framework: EO/AO papers use Boolean algebra as the bridge between logic and hardware, and questions on it stay at the level of naming a law or applying De Morgan's theorem. The examiner has made the item answerable from either arithmetic or Boolean algebra, and both routes reach the same option, which is a sign of a well-set question rather than an ambiguous one. The habit rewarded is testing a proposed identity with the smallest counterexample available instead of trying to recall which named law it is.

• An operation is commutative when swapping the operands leaves the result unchanged.
• Addition and multiplication are commutative; subtraction and division are not.
• Subtraction fails on a single counterexample — five minus three is not three minus five.
• A minus b equals b minus A only when A and b are equal, so it is an equation rather than an identity.
• Boolean algebra has three operations — complement, logical sum and logical product — and no subtraction.
• The Boolean commutative laws are stated for the sum and the product alone.
• Commutativity concerns the order of two operands; associativity concerns the grouping of three; the two are independent properties.
• Other non-commutative operations worth knowing: matrix multiplication, composition of functions, exponentiation and set difference.

• Skipping the bold-italic 'not' and marking the first true statement in the list.
• Reading the mixed upper and lower case as a mathematical distinction rather than a printing feature.
• Confusing commutativity with associativity — an operation may satisfy one and fail the other.
• Accepting an identity because it holds for one convenient pair of values; a law must hold for all of them.

Boolean algebra items in EO/AO papers ask either for the name of a law, for the result of applying De Morgan's theorem, or for the simplification of a short expression. They are quick marks for a candidate who has the law names attached to their statements, and the wrong options are typically true statements of a DIFFERENT law rather than false statements, so read the question's ask carefully before comparing the options.

No directly related past PYQ was found.

• practice — not a real PYQ

  Which one of the following operations is NOT associative ?

  • (a)Addition of real numbers
  • (b)Multiplication of real numbers
  • (c)Division of real numbers
  • (d)Union of sets

  Answer(c) Division of real numbers

• practice — not a real PYQ

  According to De Morgan's theorem, the complement of the logical sum of two variables is equal to :

  • (a)The logical sum of their complements
  • (b)The logical product of their complements
  • (c)The complement of their logical product
  • (d)The logical sum of the variables themselves

  Answer(b) The logical product of their complements