Sets, Relations, Functions & Logical Reasoning

Weightage: Roughly 20 marks — sets, relations and functions from the Business Mathematics section, plus the entire 20-mark Logical Reasoning section. Logical Reasoning alone is the cheapest block anywhere in the Foundation, and it is the one most consistently under-prepared.

Sets

A set is a well-defined collection of distinct objects. "Well-defined" means that for any object it can be determined unambiguously whether it belongs. The collection of tall people is not a set; the collection of people over 180 cm is.

Sets are written in roster form, listing the elements — $A=\{2,4,6,8\}$ — or in set-builder form, stating the defining property — $A=\{x:x\text{ is an even number},x<10\}$.

Order and repetition are irrelevant: $\{1,2,3\}$, $\{3,1,2\}$ and $\{1,2,2,3\}$ are the same set.

Types of set

• Null or empty set, written $\varnothing$ or $\{\}$, has no elements. Note that $\{0\}$ and $\{\varnothing \}$ are not empty — each contains one element.
• Singleton set has exactly one element.
• Finite and infinite sets, according to whether the elements can be counted.
• Equal sets have exactly the same elements. Equivalent sets have the same number of elements but not necessarily the same ones. Equal sets are always equivalent; the converse fails.
• Subset — $A\subseteq B$ if every element of $A$ is in $B$. A proper subset $A\subset B$ additionally requires $A\ne B$. Every set is a subset of itself, and the empty set is a subset of every set.
• Power set $P(A)$ is the set of all subsets of $A$. If $A$ has $n$ elements, $P(A)$ has $2^n$ elements, of which $2^n-1$ are proper subsets.
• Universal set $U$ contains all objects under consideration in a given context.

Operations

• Union $A\cup B$ — elements in $A$ or $B$ or both.
• Intersection $A\cap B$ — elements in both.
• Difference $A-B$ — elements in $A$ but not in $B$.
• Complement $A^{\prime }$ — elements of $U$ not in $A$.
• Symmetric difference $A△B$ — elements in exactly one of the two, that is $(A-B)\cup (B-A)$.

Disjoint sets have no common element, so $A\cap B=\varnothing$.

Laws

Union and intersection are commutative and associative, and each distributes over the other:

$A\cup (B\cap C)=(A\cup B)\cap (A\cup C),A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

De Morgan's laws are examined regularly:

$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

In words: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. The operation flips when the complement is distributed.

Cardinality

For two sets:

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

The subtraction is necessary because elements in both sets would otherwise be counted twice. This is the inclusion-exclusion principle, and it extends to three sets:

$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)$

The pattern is worth understanding rather than memorising: singles are added, pairs subtracted because they were double-counted, and the triple added back because subtracting all three pairs removed it once too often.

Venn diagram method. For any survey question, draw the diagram and fill it from the innermost region outward — start with the number in all three sets, then work out to the pairwise regions by subtraction, then to the single regions. Filling from the outside in produces errors because the outer figures usually include the inner ones.

Relations

An ordered pair $(a,b)$ has a first and second element, and $(a,b)\ne (b,a)$ unless $a=b$.

The Cartesian product $A\times B$ is the set of all ordered pairs with first element from $A$ and second from $B$. If $n(A)=p$ and $n(B)=q$, then $n(A\times B)=pq$.

A relation from $A$ to $B$ is any subset of $A\times B$. Since $A\times B$ has $pq$ elements, the number of possible relations is $2^{pq}$.

The domain is the set of first elements appearing in the relation, and the range is the set of second elements.

Types of relation

For a relation $R$ on a set $A$:

• Reflexive — $(a,a)\in R$ for every $a\in A$. Every element relates to itself.
• Symmetric — whenever $(a,b)\in R$, also $(b,a)\in R$.
• Transitive — whenever $(a,b)\in R$ and $(b,c)\in R$, also $(a,c)\in R$.
• Equivalence relation — reflexive, symmetric and transitive together.

"Is equal to" is an equivalence relation. "Is the brother of" is symmetric only in some cases and is not reflexive. "Is less than" is transitive but neither reflexive nor symmetric.

Functions

A function from $A$ to $B$ assigns to each element of $A$ exactly one element of $B$. Every function is a relation, but a relation is a function only if it satisfies both conditions: no element of the domain is left unmapped, and none maps to more than one value.

• One-one (injective) — distinct elements of the domain have distinct images.
• Onto (surjective) — every element of the codomain is the image of some element.
• Bijective — both one-one and onto. Only a bijective function has an inverse.

The composite function $(g\circ f)(x)=g(f(x))$ applies $f$ first and then $g$. Note the order: the function written on the right acts first, which is the reverse of the reading order and a standard source of error.

The inverse function $f^{-1}$ reverses the mapping, so $f^{-1}(f(x))=x$. It exists only where $f$ is bijective, because otherwise the reversal would either be undefined somewhere or ambiguous.

Logical Reasoning

This section is 20 marks — a fifth of the paper — and requires no formula, no theory and no revision. It improves purely with practice and does not decay when left for a period. Prepared properly it is close to free marks, and it is the block most consistently neglected.

Number series

A sequence follows a pattern and a term must be supplied. Work through the candidate patterns in order:

1. Constant difference — arithmetic progression.
2. Constant ratio — geometric progression.
3. Changing difference — compute the differences, then the differences of those differences.
4. Squares, cubes or their neighbours — 2, 5, 10, 17 is $n^2+1$.
5. Alternating series — two interleaved sequences, which is why terms should be examined in alternate positions when no simple pattern appears.
6. Operations combined — multiply then add, as in 3, 7, 15, 31, where each term is twice the previous plus one.

Compute the first differences before anything else; they resolve most series immediately.

Coding and decoding

A word is encoded by a rule and another must be encoded or decoded. The common rules:

• Letter shift — each letter moves forward or back by a fixed number. Knowing the alphabet positions is essential, and it is worth memorising that A is 1, E is 5, J is 10, O is 15, T is 20 and Z is 26.
• Reversal — the word is written backwards, possibly with a shift.
• Positional substitution — letters replaced by their positions or by the letter at the mirror position, where A pairs with Z, B with Y, and so on.
• Number coding — each word is assigned a number by a rule involving letter count or letter values.

Write out the letters with their positions beneath. The rule almost always becomes visible once the numbers are written down.

Odd man out

One item differs from the rest by a property. Test in this order: type or category; a numerical property such as prime, perfect square or divisibility; a structural property such as the number of letters or vowels; and finally the relationship between paired items.

Direction sense

A person moves through a series of turns and the final position or displacement must be found.

Always draw the diagram, fixing north at the top. Mark each movement in sequence. Where a distance is asked, the answer is usually the hypotenuse of a right triangle formed by the net eastward and net northward displacements, obtained by Pythagoras.

Remember that a right turn is clockwise and a left turn anticlockwise from the direction currently faced, not from north.

Seating arrangements

Persons are placed in a row or around a table subject to conditions.

The method: draw the arrangement, place the most constrained person first — the one whose position is fixed absolutely, such as "P sits at the extreme left" — and then place others relative to them. Where a condition permits two possibilities, draw both cases and eliminate one using a later condition.

For circular arrangements, establish whether the persons face the centre or outward, because left and right reverse between the two. This detail decides many questions and is easily missed.

Blood relations

Draw a family tree. Use a consistent notation: a horizontal line for a marriage, a vertical line for a parent-child relationship, and a symbol for sex where the question distinguishes it.

Work through the statement one relationship at a time rather than trying to hold it in mind. In puzzles phrased as "pointing to a photograph", identify who the speaker is first, then trace the chain from them.

Syllogisms

Two or more statements are given and it must be determined which conclusions follow necessarily.

The essential discipline is to accept the statements as true even where they contradict common knowledge, and to ask only whether the conclusion follows of necessity — not whether it might be true.

Draw Venn diagrams. Where a statement can be represented in more than one way, draw every possibility, and accept a conclusion only if it holds in all of them.

The standard forms:

• All A are B — circle A entirely inside circle B.
• No A is B — two disjoint circles.
• Some A are B — overlapping circles.
• Some A are not B — part of A lies outside B.

The most examined trap: from "All A are B" it does not follow that "All B are A". From "Some A are B" it does follow that "Some B are A", because overlap is symmetric.

How this chapter is examined

Sets appear as a survey problem requiring the inclusion-exclusion formula or a Venn diagram, as a question on the number of subsets or elements of a power set, or as an application of De Morgan's laws. Relations and functions appear as identification questions — is this relation reflexive, is this function one-one — and as a composite or inverse function computation.

Logical Reasoning appears as a spread across all its types, and it is answered by drawing rather than by reasoning in the head. The single most valuable habit in the whole section is to draw the diagram — the series differences, the alphabet positions, the direction path, the seating circle, the family tree, the Venn diagram — before attempting any conclusion.