Set, Logic and Functions

This is the smallest chapter of the paper (our estimate: about 3 questions, mostly Section A), but it is also the vocabulary of every other chapter: you cannot find a domain, an inverse or a range in calculus without it. The questions are short and exact: count something with a Venn diagram, solve an absolute-value inequality, negate a statement, decide whether a function is one-one or onto, or compose two functions. Marks are lost here by carelessness, not by difficulty.

1.1 Set, real number system, intervals, absolute value, logic, connectives, laws of logic

Sets

A set is a well-defined collection. A\subseteq B means every element of A is in B; A\subset B (proper) also needs A\ne B. The power set P(A) is the set of all subsets, so n(P(A))=2^{n} when n(A)=n; there are 2^n-1 proper subsets and 2^n-2 non-empty proper subsets.

Operation	Symbol	Meaning
union	A\cup B	in A or B or both
intersection	A\cap B	in both
difference	A-B	in A but not B
complement	A'	in U but not A
symmetric difference	A\,\Delta\,B	(A-B)\cup(B-A)

Laws. Commutative, associative, distributive (A\cap(B\cup C)=(A\cap B)\cup(A\cap C)), and De Morgan: (A\cup B)'=A'\cap B', (A\cap B)'=A'\cup B'; also A-B=A\cap B'.

Counting. n(A\cup B)=n(A)+n(B)-n(A\cap B) and

Number of elements in exactly one of A,B: n(A)+n(B)-2n(A\cap B).

Real number system and intervals

\mathrm{N}\subset\mathrm{W}\subset\mathrm{Z}\subset\mathrm{Q}\subset\mathrm{R}. Rationals have terminating or repeating decimals; irrationals (\sqrt{2}, \pi, e) have non-terminating, non-repeating ones. Between any two reals there are infinitely many rationals and irrationals.

Intervals: (a,b) open, [a,b] closed, [a,b) and (a,b] half-open; (-\infty,a), [a,\infty) are unbounded. Union and intersection of intervals are the usual set operations on a number line.

Absolute value

|x|=x if x\ge0 and -x if x<0; |x|=\sqrt{x^2}.

Inequality (r>0)	Solution
|x-a|<r	a-r<x<a+r
|x-a|\le r	a-r\le x\le a+r
|x-a|>r	x<a-r or x>a+r
|f|<|g|	square both sides: f^2<g^2

Triangle inequality: |a+b|\le|a|+|b|; |ab|=|a||b|.

Logic

A statement is a sentence that is either true or false. Connectives: \neg p (not), p\wedge q (and), p\vee q (or, inclusive), p\to q (if p then q), p\leftrightarrow q (if and only if).

p	q	p\wedge q	p\vee q	p\to q	p\leftrightarrow q
T	T	T	T	T	T
T	F	F	T	F	F
F	T	F	T	T	F
F	F	F	F	T	T

p\to q is false only when p is true and q false, and p\to q\equiv\neg p\vee q.

For p\to q: converse q\to p; inverse \neg p\to\neg q; contrapositive \neg q\to\neg p. The contrapositive is equivalent to the original; the converse and inverse are equivalent to each other but not to it.

Laws of logic. De Morgan: \neg(p\wedge q)\equiv\neg p\vee\neg q, \neg(p\vee q)\equiv\neg p\wedge\neg q; distributive, idempotent, double negation \neg\neg p\equiv p; \neg(p\to q)\equiv p\wedge\neg q. A tautology is always true (e.g. p\vee\neg p, p\to(p\vee q)); a contradiction is always false (p\wedge\neg p). Quantifiers: the negation of "for all x, P(x)" is "there exists x with \neg P(x)", and vice versa.

1.2 Function, types of functions, inverse and composite functions

A function f:A\to B gives each x\in A exactly one f(x)\in B. A is the domain, B the codomain, and \{f(x)\} the range (a subset of B). A vertical line meets a graph at most once.

Finding a domain: denominators \ne0; even roots need radicand \ge0; \log needs argument >0 (and base >0,\ne1); \sin^{-1}x,\cos^{-1}x need |x|\le1.

One-one, onto, bijective

Type	Test	Counting (n(A)=m, n(B)=n)
injective (one-one)	f(a)=f(b)\Rightarrow a=b	n(n-1)\cdots(n-m+1)
surjective (onto)	range = codomain	n^m-\binom{n}{1}(n-1)^m+\binom{n}{2}(n-2)^m-\cdots
bijective	both	n! if m=n
all functions		n^m

A graph passing the horizontal line test is injective. A function from \mathrm{R} to \mathrm{R} like x^2 is neither (not injective: f(2)=f(-2); not onto: negatives missed); x^3 is bijective.

Standard families

Algebraic (polynomial, rational, root); trigonometric; exponential a^x (a>0, range (0,\infty)); logarithmic \log_a x (domain (0,\infty), inverse of a^x). Log laws: \log(xy)=\log x+\log y, \log(x/y)=\log x-\log y, \log x^k=k\log x, \log_a b=\dfrac{\log b}{\log a}.

Even / odd. Even: f(-x)=f(x) (graph symmetric about the y-axis; x^2,\cos x,|x|). Odd: f(-x)=-f(x) (x^3,\sin x,\tan x). Periodic: f(x+T)=f(x); \sin ax,\cos ax have period 2\pi/|a| and \tan ax has \pi/|a|.

Composite and inverse

(f\circ g)(x)=f(g(x)); in general f\circ g\ne g\circ f, but composition is associative. The inverse f^{-1} exists exactly when f is a bijection; to find it solve y=f(x) for x and swap. Then f(f^{-1}(x))=x, (f\circ g)^{-1}=g^{-1}\circ f^{-1}, and the graphs of f,f^{-1} are mirror images in y=x.

Formula and fact sheet

Topic	Result
subsets	2^n; proper 2^n-1
union	n(A\cup B)=n(A)+n(B)-n(A\cap B)
De Morgan	(A\cup B)'=A'\cap B'
|x|<r	-r<x<r
p\to q	\equiv\neg p\vee q; contrapositive \neg q\to\neg p
number of functions	n^m; injective ^nP_m; bijections n!
inverse	(f\circ g)^{-1}=g^{-1}\circ f^{-1}
period	\sin ax: 2\pi/|a|; \tan ax: \pi/|a|

Chapter summary

• A set of n elements has 2^n subsets, 2^n-1 proper ones.
• Use inclusion-exclusion; keep the three-set formula's alternating signs.
• |x-a|<r is an interval around a; >r is its outside.
• p\to q is false only for T → F; the contrapositive is equivalent, the converse is not.
• Injective means distinct inputs give distinct outputs; onto means range = codomain.
• Inverse exists only for bijections; compose in reverse order to invert a composite.

Practice questions

Set A — Recall (14 questions)

1. The number of subsets of a set with 4 elements isa8b15c16d12
2. Which of the following is irrational?a0.1010010001\ldots (the number of zeros grows by one each time)b22/7c\sqrt{49}d3.14
3. The solution set of |x-3|<2 isa(-1,5)b(1,5)c(-\infty,1)\cup(5,\infty)d[1,5]
4. The negation of p\wedge q isa\neg p\wedge\neg qbp\vee qc\neg p\wedge qd\neg p\vee\neg q
5. The contrapositive of p\to q isa\neg q\to\neg pbq\to pc\neg p\to\neg qdp\to\neg q
6. If n(A)=20, n(B)=15 and n(A\cap B)=7, then n(A\cup B) isa35b28c42d22
7. The number of injective functions from a 3-element set to a 4-element set isa64b12c81d24
8. The domain of f(x)=\sqrt{x-2}+\dfrac{1}{x-5} isa[2,\infty)b(2,5)\cup(5,\infty)c[2,5)\cup(5,\infty)d\mathrm{R}-\{5\}
9. The inverse of f(x)=2x+3 isa\dfrac{x+3}{2}b\dfrac{x-3}{2}c2x-3d\dfrac{1}{2x+3}
10. If f(x)=x^2 and g(x)=x+1, then (f\circ g)(3) isa16b10c12d9
11. Which function is even?ax^3+xb\cos x+x^2c\sin xdx^2+x
12. The period of \sin 3x isa2\pib\pic\pi/3d2\pi/3
13. The value of \log_2 8+\log_3 9 isa5b6c4d11
14. The function f:\mathrm{R}\to\mathrm{R}, f(x)=x^2 isaone-one onlybonto onlycneither one-one nor ontodbijective

Answers and explanations — Set A

1. (c) — 2^4=16. (b) 15 is the number of proper subsets, 2^4-1.

2. (a) — the decimal never terminates or repeats. 22/7 is a fraction, \sqrt{49}=7, 3.14 terminates.

3. (b) — 3-2<x<3+2. (c) is the answer to |x-3|>2.

4. (d) — De Morgan: \neg(p\wedge q)\equiv\neg p\vee\neg q.

5. (a) — contrapositive swaps and negates. (b) is the converse, (c) the inverse.

6. (b) — 20+15-7=28.

7. (d) — 4\times3\times2=24 (4^3=64 counts all functions).

8. (c) — x\ge2 for the root, x\ne5 for the fraction; 2 is allowed because the root of 0 is defined.

9. (b) — y=2x+3\Rightarrow x=(y-3)/2.

10. (a) — g(3)=4, f(4)=16. (b) 10 is (g\circ f)(3).

11. (b) — \cos(-x)+(-x)^2=\cos x+x^2. x^3+x and \sin x are odd; x^2+x is neither.

12. (d) — 2\pi/3.

13. (a) — 3+2.

14. (c) — f(2)=f(-2) and no real x gives -1.

Set B — Understanding (9 questions)

1. In a group of 60 people, 35 like tea, 30 like coffee and 10 like both. The number who like neither isa0b10c5d15
2. The number of integers x with |2x-1|\le5 isa5b6c7d3
3. The solution of |x-2|>|x+1| isax<\tfrac{1}{2}bx>\tfrac{1}{2}cx<-1devery real x
4. p\to q is logically equivalent toap\wedge\neg qb\neg p\vee qcp\vee\neg qd\neg p\wedge q
5. Which is a tautology?ap\wedge\neg pbp\wedge qcp\to(p\vee q)d(p\wedge q)\to\neg p
6. The number of subsets of \{1,2,3,4,5\} that contain both 1 and 2 isa4b6c16d8
7. If f(x)=\dfrac{x+1}{x-1} (x\ne1), then f(f(x)) equalsaxb1/xc-xd\dfrac{x-1}{x+1}
8. If f(x)=2x+1 and g(x)=x-3, then (g\circ f)^{-1}(5) isa4b7c3.5d2
9. The range of f(x)=\dfrac{1}{1+x^2} isa(0,1)b[0,1]c(0,\infty)d(0,1]

Answers and explanations — Set B

1. (c) — 60-(35+30-10)=5.

2. (b) — -5\le2x-1\le5\Rightarrow-2\le x\le3: the integers -2,-1,0,1,2,3.

3. (a) — square both sides: x^2-4x+4>x^2+2x+1\Rightarrow 3>6x. Check x=0: 2>1 ✓.

4. (b) — the implication fails only when p true and q false, so it equals \neg p\vee q (p\wedge\neg q is its negation).

5. (c) — if p is true then p\vee q is true. (d) fails when p,q are both true.

6. (d) — fix 1 and 2 in; each of 3, 4, 5 is in or out: 2^3=8.

7. (a) — \dfrac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=\dfrac{2x}{2}=x; f is its own inverse.

8. (c) — (g\circ f)(x)=2x-2, so the inverse is \dfrac{y+2}{2}; at 5 it is 3.5.

9. (d) — x=0 gives 1; the value never reaches 0. (a) wrongly drops 1.

Set C — Application (7 questions)

1. Among 100 students, 50 study Maths, 40 Physics, 30 Chemistry; 20 study Maths and Physics, 15 Physics and Chemistry, 10 Maths and Chemistry, and 5 study all three. How many study at least one subject?a80b85c70d75
2. The domain of f(x)=\log(x^2-5x+6) isa(2,3)b(-\infty,2)\cup(3,\infty)c[2,3]d\mathrm{R}
3. The function f(x)=\ln\!\big(x+\sqrt{x^2+1}\big) isaevenbperiodiccodddneither even nor odd
4. The solution set of |x^2-4|<5 isa(-1,3)b(-\sqrt{5},\sqrt{5})c(-9,9)d(-3,3)
5. If f(x+1)=x^2+3x+2, then f(x) isax^2-xbx^2+xcx^2+3xdx^2+1
6. The number of onto functions from \{1,2,3\} to \{a,b\} isa8b2c6d4
7. If A=\{1,2\}, then the number of elements of P(P(A)) isa16b8c4d32

Answers and explanations — Set C

1. (a) — 50+40+30-20-15-10+5=80.

2. (b) — x^2-5x+6=(x-2)(x-3)>0 outside the roots.

3. (c) — f(-x)=\ln(\sqrt{x^2+1}-x)=\ln\frac{1}{x+\sqrt{x^2+1}}=-f(x).

4. (d) — -5<x^2-4<5\Rightarrow x^2<9 (the left side x^2>-1 always holds), so -3<x<3.

5. (b) — put x\to x-1: (x-1)^2+3(x-1)+2=x^2+x.

6. (c) — all functions 2^3=8, minus the 2 constant ones: 6.

7. (a) — n(P(A))=4, so n(P(P(A)))=2^4=16.