Binary Operations

1. Binary Operations
2021

2. Binary Operations
Take the multiplication
operation on the set of integers,
Z = {0, ±1, ±2, ±3,…}:
x = +3 and y = +4
x × y = +12
(x, y) ∈ Z × Z and x × y ∈ Z
We write *: Z × Z → Z; (x, y) → x × y

3. Binary Operations
Take the addition operation
on the set of integers, Z =
{0, ±1, ±2, ±3,…}:
x = +3 and y = -4
x + y = +3 + (-4) = -1∈ Z
(x, y) ∈ Z × Z and x + y ∈ Z
We write *: Z × Z → Z; (x, y) → x + y

4. Binary Operation:
: (x, y ) → f(x, y)
That is a calculation that combines two elements
of the set (called operands) to produce another
element of the set. It maps elements of the
Cartesian product A × A into A itself. i.e. f: A × A
→ A. We call the operation multiplication!

5. Binary Operation notation
• We usually write binary operations
infix that is, between their arguments,
rather than prefix, before their
arguments.
• Also, the symbols chosen for binary
operations are usually reminiscent of
+ or × rather than letters.

6. Examples: Infix notation
Note the infix positioning of the operation
characteristic of binary operations.

7. Binary operations on Z
Given the set of integers X, we normally call the
operation “multiplication” and denote it by * even
though we can define the operations in many ways:
• x * y = x + 2y + 3
• x * y = 1 + xy
• x * y = 3x + y
or even
• x * y = x + y

8. Binary Operation Examples

9. Binary Operations Examples

10. Binary Operations Examples

11. Are these binary operations?
1. a * b = a + b – 1
2. a * b = 1 – 2ab
3. a * b = a/b
4. a * b = a/b on Z {0}
5. For a, b ∈ A, a * b = a + b where A is the set A
= {1, 2, 3, 4}.
Given a, b ∈ Z. Consider the operations *: Z × Z → Z
as follows:

12. Example: Closure

13. Some interesting questions?
• Commutativity: does the order in which we
perform the operation matter?
• Associativity: does it matter how we group the
operands?
• Identity: is there are element which when used
as an operand with another object returns the
same object?

14. Binary
Operati
on
Definitions:

15. Addition: +: R × R → R; (x, y) → x + y

16. Multiplication:*: R × R → R; (x, y)→ x × y

17. Multiplication:*: R × R → R; (x, y)→ x × y

18. Subtraction: *: R × R → R; (x, y) → x - y

19. Subtraction: *: R × R → R; (x, y) → x - y

20. Division: *: R{0} × R{0} → R{0}; (x, y) → x / y

21. Division: *: R{0} × R{0} → R{0}; (x, y) → x / y

22. Define an identity element for a binary operation.

23. Binary
Operati
on
Identity Element

24. RTP: The Identity Element is Unique

25. The identity element e of a binary
operation  is unique.
Suppose that e and e’ are two
identities of *.
Then, e * e’ = e since e’ is an
identity and also e * e’ = e’ since e
is an identity and hence e = e’ and
the identities are unique. ■

26. Identity Element

27. Is there an identity element for
this binary operation?
• Let * be the binary operation on S
= {1, -1, i, -i} with i2 = -1 defined
by a * b = a x b.
• Is there an identity element for *?
* 1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1

28. Binary
Operati
on
Identity Element
• For the set Z, the integer 1 is an identity w. r. t.
“”, but not w. r. t. “+”.
• The number 0 is an identity w. r. t. “+”.

29. Binary
Operati
on
Identity Element
Let “” be the operation defined on Z, x  y = x +
y  1. Then is there an identity?

30. Identity Element

31. Let “∆” be the operation defined on Z, x ∆ y = 1 + xy.
Then the operation has no identity element in Z
Hint: Assume there is an identity element e, then
x ∆ e = 1 + xe = x
What does that give us?

32. SOLUTION
Let “∆” be the operation defined on Z, x ∆ y = 1 + xy.
Then the operation has no identity element in Z.
SOLUTION
Assume there is an identity element e, then
x ∆ e = 1 + xe = x
⇒ xe = x - 1
⇒ e = (x – 1)/x = 1 - 1/x ∉ Z

33. A binary operation which is NOT commutative?

34. A binary operation which is NOT associative?

35. Explanation

37. Using a table to represent a Binary operation
• Consider the set A = {-1, 1} under the usual
multiplication as we know it. We can think of x, and y
as the elements of this set. Let us compute all the
products for the elements of this set.
• This produces the results:
1 * 1 = 1 × 1 = 1
1 * -1 = 1 × -1 = -1
-1 * 1 = -1 × 1 = -1
-1 * -1 = -1 × -1 = 1

38. Summarising, in a table we have…
• Consider the set A = {-1, 1} under the usual
multiplication as we know it. We can think of x, and y
as the elements of this set. Let us compute all the
products for the elements of this set.
• This produces the results:
1 * 1 = 1 × 1 = 1
1 * -1 = 1 × -1 = -1
-1 * 1 = -1 × 1 = -1
-1 * -1 = -1 × -1 = 1
* 1 -1
1 1 -1
-1 -1 1

39. Cayley Table
When we are given a binary
operation on a finite set, it is common
to specify it in tabular form,
sometimes called a Cayley table.

40. Arthur Cayley
Born: 16 Aug 1821 Died: 26 Jan 1895

41. • In 1863 Cayley was appointed Sadleirian
professor of Pure Mathematics at Cambridge.
• He published over 900 papers and notes
covering nearly every aspect of modern
mathematics.
Arthur Cayley

42. The most important of his work was
developing the algebra of matrices, work in
non-Euclidean geometry and n-dimensional
geometry.
As early as 1849 Cayley wrote a paper
linking his ideas on permutations with
Cauchy's.
 In 1854 Cayley wrote two papers which are
remarkable for the insight they have of
abstract groups.
Arthur Cayley

43. At that time the only known groups were
permutation groups and even this was a
radically new area, yet Cayley defines an
abstract group and gives a table to display the
group multiplication.
These tables become known as Cayley Tables.
Arthur Cayley

44. He gives the 'Cayley tables' of some special
permutation groups but, much more
significantly for the introduction of the
abstract group concept, he realised that
matrices were groups .
Arthur Cayley

45. Cayley Table
• Take * to be the binary
operation on S = {1, -1, i, -
i} with i2 = -1 defined by a *
b = a × b the corresponding
Cayley table is:
• Here is another example of a Cayley table.
* 1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1

46. Let “” be the operation defined as x  y = x + y  1 on
Z, show that “” is both commutative and associative.

47. SOLUTION
Let “” be the operation defined as x  y = x + y  1 on Z,
then “” is both commutative and associative.
Proof: Commutative:
x * y = x + y – 1 = y + x – 1 = y * x
Associative:
x * (y * z) = x * (y + z – 1)
= x + (y + z – 1) -1
= (x + y – 1) * z = (x * y) * z
Thus, * is both commutative and associative on Z.

48. Let *: Z × Z → Z; (x, y) →1 + xy. Show
that * is commutative but not associative.

49. Show: x  y = x + 2y. This operation is neither
commutative, nor associative.

50. Show: x  y = x + 2y. This operation is neither
commutative, nor associative.
• Commutative:
x * y = x + 2y ≠ y + 2x = y * x
So, it is not commutative.
• Associative:
x * (y * z) = x * (y + 2z)
= x + 2(y + 2z) = x + 2y + 4z
(x * y) * z = (x + 2y) * z = x + 2y + 2z
So, * is not associative

51. Consider the Binary Operation * as given by…
Find the answer to:
1) a * (b * c)
2) (a * b) * c
3) a * (b * (c * d))

52. Consider the Binary Operation * as given by…
On what set is the operation *
defined?
Is the operation * closed?
Is there an element e such
that a * e = a = e * a?

53. Inverses

54. Inverses
• Is this a binary operation?
• Is there an identity element?
• Is there an element x-1 such
that for every x we have x *
x-1 = e = x-1 * x?

55. Let * be an associative binary operation on a set A with identity
e. If x has an inverse, show that this inverse is unique.
Let y and z be inverses of x then
Hence y = z.
Thus, if x has an inverse, that inverse is unique.

56. Let * be an associative binary operation on a set A with identity
e. If x has an inverse, show that this inverse is unique.
Let y and z be inverses of x then
y = y * e By the property of identity
= y * (x * z) Since z is an inverse of x
= (y * x) * z Since * is associative
= e * z Since y is an inverse of x
= z By the property of identity
Hence y = z.
Thus, if x has an inverse, that inverse is unique.

57. Inverses

58. 2 x 2 Matrices with real coefficients
under component wise addition
Consider the set of 2 x 2 matrices with real
coefficients under component wise addition…
• Is this a binary operation?
• Is the operation commutative and associative?
• What is the identity for the operation?
• Do the elements have inverses?

59. 2 x 2 Matrices with real coefficients
under component wise addition
Consider the set of 2 x 2 matrices with real
coefficients under component wise addition…

60. Do inverses exist?
• Consider the set of 2 x 2 matrices over the field
of real numbers. Is multiplication a binary
operation?
• Is there an identity element?
• Do all 2 x 2 matrices have inverses?

61. Do inverses exist?
• Recall that the inverse of a matrix is A-1 = adj
A/det A. i.e. the matrix of cofactors divided by
the determinant of the matrix.
• For a 2 x 2 matrix to have an inverse, we must
have det A ≠ 0.
• Thus all 2 x 2 matrices do not have inverses
under multiplication.
• Thus, all elements are not invertible.

64. Complex Numbers
• Consider * to be the binary
operation on S = {1, -1, i, -i} with i2
= -1 defined by a * b = a x b.
• The corresponding Cayley table is:
• Is this binary operation closed,
commutative, associative?
• Is there an identity?
• Are there inverses?
* 1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1

65. Is there a way we can check for commutativity
quickly from the Cayley table?…

66. How would you check for associativity?…

67. Closed?

68. Commutative?

69. Associative?

70. Identity element?

71. Example – Power Set
• The intersection and union of two subsets of a set A are
both binary operations on the power set of A, P(A).
• If A = {1, 2} what would be the identity element for
each of these operations?

72. Example – Power Set
• The intersection and union of two subsets of a set A are both
binary operations on the power set of A, P(A).
• If A = {1, 2} what would be the identity element for each of these
operations?
SOLUTION
A= {1, 2} and P(A) = {{}, {1}, {2}, {1, 2}}
• Intersection: ∩: P(A) × P(A) → P(A)
– {1} ∩ {1, 2} = {1} ∈ P(A)

73. Symmetric Difference

75. Symmetric Difference

76. Does this remind us of the Exclusive OR in
propositional logic?

78. Given sets A, B and C. Show that ∆ is associative…
Similarly,

81. Some Past Paper Questions

82. Given a binary operation x  y = x + 2y.This
operation has no identity, thus no inverse.
Try…
x * e =
e * x =
Do you get two different values for the identity?

83. Given a binary operation x  y = x + 2y.This
operation has no identity, thus no inverse.
Consider:
x * e = x + 2e = x ⇒ e = 0
e * x = e + 2x = x ⇒ e = -x
But we know that the identity is unique. Thus, the
operation * has no identity and hence no inverse.

84. Question

85. Question May 2017
a) Commutative:
• 3 * 4 = 3 × 3 + 4 × 4 – 3 × 4 = 25 – 12 = 13
• 4 * 3 = 3 × 4 + 4 × 3 – 4 × 3 = 24 – 12 = 12
So it is not commutative.
b) Identity: If there is an identity e, then n * e = n = e * n ⇒ 3n + 4e – ne = n
then 2n = e(4 – n) and e = 2n/(4 – n). Now for n = 1 e = 2/3 ∉ Z and hence this
operation does not have an identity element. Also 4e + 3n – en = 3e + 4n – ne
So 3n + 4e = 3e + 4n and n = e but the identity is unique so there is no identity.
c) Inverse: Since it does not have an identity, it has no inverses.

86. Past Paper Question

87. Questions