Relations and Functions

Maths · Class 11

Lesson 10 of 10 · 15 min

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Must-know facts

18 facts

  1. 1(a, b) = (c, d) exactly when a = c and b = d.
  2. 2A × B = {(a, b) : a ∈ A, b ∈ B}; A × φ = φ.
  3. 3n(A × B) = n(A) · n(B).
  4. 4A × B ≠ B × A in general, but the two have the same number of elements.
  5. 5A × (B ∩ C) = (A × B) ∩ (A × C) and A × (B ∪ C) = (A × B) ∪ (A × C).
  6. 6R × R is the plane; A × A × A is a set of ordered triplets.
  7. 7A relation from A to B is any subset of A × B; there are 2^(pq) of them when n(A) = p and n(B) = q.
  8. 8Domain: first entries. Range: second entries. Codomain: B. Range ⊂ codomain.
  9. 9A function gives every element of its domain exactly one image; no two pairs share a first entry.
  10. 10There are qᵖ functions from a set of p elements to a set of q elements.
  11. 11Real valued function: range in R. Real function: domain and range both in R.
  12. 12Identity: domain R, range R. Constant c: domain R, range {c}.
  13. 13x²: range [0, ∞). x³: range R. 1/x: domain and range R − {0}.
  14. 14|x|: domain R, range [0, ∞); V-shaped graph with vertex at the origin.
  15. 15Signum: domain R, range {−1, 0, 1}; sgn 0 = 0.
  16. 16[x] = n ⇔ n ≤ x < n + 1; range Z; [−2.4] = −3.
  17. 17Natural domain: denominator ≠ 0 and expression under √ ≥ 0.
  18. 18f/g is defined only where g(x) ≠ 0.

Common traps

Where marks are lost

Writing A × B = B × A because both have pq elements.

Equal counts are not equal sets. (a, b) ∈ A × B but (b, a) need not be, so A × B ≠ B × A unless A = B (or one set is empty).

Counting 2^(p + q) or pq relations from A to B.

A × B has pq pairs and a relation is any subset of it, so there are 2^(pq) relations. For p = 2, q = 4 that is 2⁸ = 256.

Calling a relation a function when some element of A has no image.

A function needs an image for every element of A. An element left out of the domain breaks the rule just as an element with two images does.

Rejecting a function because two elements share an image.

Many-to-one is allowed: f(x) = x² gives f(2) = f(−2) = 4 and is still a function. Only one element sending two arrows is forbidden.

Giving the codomain when the range is asked.

The range is the set of images actually reached. For f : R → R, f(x) = x², the codomain is R but the range is [0, ∞).

Writing [−2.4] = −2 by dropping the decimal part.

[x] is the greatest integer not exceeding x. −2 > −2.4, so it is too big; the answer is −3. Only for x ≥ 0 does [x] simply drop the decimal part.

Giving 1 or −1 as the value of the signum function at 0.

sgn 0 = 0, so the range is {−1, 0, 1}. The formula |x|/x matches the signum function only for x ≠ 0.

Finding the domain of f/g from the formula of f alone.

Remove every x with g(x) = 0, and keep only the points where both f and g are defined.

Formulas

10 to know

Cartesian product

A × B = {(a, b) : a ∈ A, b ∈ B}

A × φ = φ; ordered pairs, so order matters.

Size of a product

n(A × B) = n(A) · n(B)

If n(A) = p and n(B) = q, A × B has pq pairs.

Number of relations

2^(pq)

Relations from A to B are the subsets of A × B.

Number of functions

qᵖ

Each of the p elements of A picks one of the q elements of B.

Identity and constant

f(x) = x; f(x) = c

Ranges R and {c}.

Modulus function

|x| = x if x ≥ 0; |x| = −x if x < 0

Range [0, ∞).

Signum function

sgn x = 1 if x > 0; 0 if x = 0; −1 if x < 0

Range {−1, 0, 1}; equals |x|/x for x ≠ 0.

Greatest integer function

[x] = n ⇔ n ≤ x < n + 1, n ∈ Z

Range Z; [x] ≤ x < [x] + 1.

Sum, difference, scalar multiple

(f ± g)(x) = f(x) ± g(x); (αf)(x) = α f(x)

On the common domain.

Product and quotient

(fg)(x) = f(x) g(x); (f/g)(x) = f(x)/g(x)

The quotient needs g(x) ≠ 0.

Key terms

16 terms

Ordered pair
Two objects in a fixed order, (a, b), where swapping them gives a different pair.
Cartesian product
A × B, the set of all ordered pairs with first entry from A and second from B.
Ordered triplet
Three objects in a fixed order, (a, b, c); A × A × A is a set of them.
Relation
Any subset of A × B, picked out by a rule linking first and second entries.
Image
For a pair (x, y) in a relation or function, y is the image of x.
Preimage
For f(x) = y, x is a preimage of y.
Domain
The set of first entries; for a function f : A → B, the whole of A.
Codomain
The set B that images are allowed to come from.
Range
The set of images actually produced; always a subset of the codomain.
Function
A relation giving every element of its domain exactly one image.
Real function
A function whose domain and range are both subsets of R.
Polynomial function
a₀ + a₁x + … + aₙxⁿ with real coefficients and n a non-negative integer.
Rational function
A quotient f(x)/g(x) of polynomial functions, defined where g(x) ≠ 0.
Modulus function
|x|, the distance of x from 0: x for x ≥ 0 and −x for x < 0.
Signum function
Gives 1, 0 or −1 according as x is positive, zero or negative.
Greatest integer function
[x], the largest integer that does not exceed x.
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