ISTANBUL AYDIN UNIVERSITY
FACULTY OF ENGINEERING
ELECTRICAL & ELECTRONICS ENGINEERING DEPT.
EEE247: DISCRETE MATHEMATICS
6. Functions
Instructor:
Assist. Prof. Dr. Mohammed ALKRUNZ
Electrical & Electronics Engineering Dept.
Functions,,,
 From calculus, you are familiar with the concept of a real-valued function
“𝑓”, which assigns to each number 𝑥 ∈ ℝ a particular value 𝑦 = 𝑓(𝑥),
where 𝑦 ∈ ℝ.
 But, the notion of a function can also be naturally generalized to the
concept of assigning elements of any set to elements of any set.
Formal Definition:
 For any set, A, B, we say that a function 𝑓 from (or “mapping”) A to B
(𝑓. 𝐴 → 𝐵) is a particular assignment of exactly one element 𝑓(𝑥) ∈ 𝐵 to
each element 𝑥 ∈ 𝐴.
 Two inputs could give one output such as 𝑦 = 𝑥 2 , but it is not possible for
one input to give two outputs. (i.e., two locations at one time)
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Graphical Representations:
 Functions can be represented graphically in several ways:
𝒇
a
𝑓 𝑎 =𝑏
𝒇
A
B
“Venn Diagram”
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“Bipartite Graph”
Dr. M. Alkrunz
EEE247: Discrete Mathematics
“Plot”
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Functions,,,
Functions that we’ve seen so far:
 A proposition can be viewed as a function from “situation” to truth values.
 A logic system called “Situation Theory”.
 𝑝 ≡ "𝑖𝑡 𝑖𝑠 𝑟𝑎𝑖𝑛𝑖𝑛𝑔“, s ≡ "𝑜𝑢𝑟 𝑠𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛 ℎ𝑒𝑟𝑒, 𝑛𝑜𝑤“
 𝑝 𝑠 ∈ 𝑇, 𝐹
 A propositional operator can be viewed as a function from ordered pairs of
truth values to truth values:
𝒙
 ∨
𝐹, 𝑇
 →
𝑇, 𝐹
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=𝑇
=𝐹
𝑻
𝑭
Dr. M. Alkrunz
EEE247: Discrete Mathematics
𝑓(𝑥)
∨
𝒚
𝑻
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Functions,,,
Functions that we’ve seen so far:
 A predicate can be viewed as a function from objects to propositions (or
truth values):
 𝑝 ≡ "𝑖𝑠 7 𝑓𝑒𝑒𝑡 𝑡𝑎𝑙𝑙“,
 𝑝 𝑀𝑖𝑘𝑒 ≡ "𝑀𝑖𝑘𝑒 𝑖𝑠 7 𝑓𝑒𝑒𝑡 𝑡𝑎𝑙𝑙“ ≡ 𝐹𝑎𝑙𝑠𝑒
 A bit string of length “n” can be viewed as a function from the numbers
1, 2, … , 𝑛 (bit positions) to the bits 0, 1
 𝐵 = 101, 𝑡ℎ𝑒𝑛 𝐵 3 = 1
 A set "𝑆“ over universe “𝑈” can be viewed as a function from the elements
of “𝑈” to 𝑇, 𝐹
 𝑆= 3,
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𝑆 0 = 𝐹,
𝑆 3 =𝑇
Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Functions that we’ve seen so far:
 A set operator such as “⋃, ⋂, −” can be viewed as a function from pairs of
sets to sets.
⋂
1, 3 , 3, 4
= 3
Some Functions Terminology:
 If 𝑓. 𝐴 → 𝐵 and 𝑓 𝑎 = 𝑏, where (𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵), then:




“A” is the domain of 𝑓.
“B” is the codomain of 𝑓.
“b” is the image of “a” under 𝑓.
“a” is a pre-image of “b” under 𝑓.
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
In general, “b” may have
more than 1 pre-image
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Functions,,,
Range V.S. Codomain:
 Suppose I declare to you that: “𝑓 is a function mapping students in this class
to the set of grades 𝐴, 𝐵, 𝐶, 𝐷, 𝐹 ”
 At this point (without exam), the codomain of 𝑓 is: 𝑨, 𝑩, 𝑪, 𝑫, 𝑭 , and
its range is: unknown.
 Suppose that grades turns out all A’s and B’s, then the range of 𝑓 will be
𝐴, 𝐵 , but its codomain is: 𝐴, 𝐵, 𝐶, 𝐷, 𝐹 .
 On other hand, the range is the actual output.
 The range: 𝑅 ⊆ 𝐵 of 𝑓 is 𝑏 ∃𝑎 𝑓 𝑎 = 𝑏
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Constructing Function Operators
 If • (“dot”) is any operator over B, then we extend • to also denote an
operator over functions 𝑓. 𝐴 → 𝐵.
 E.g., Given any binary operator •: 𝐵 × 𝐵 → 𝐵, and functions 𝑓, 𝑔. 𝐴 → 𝐵, we
define 𝑓•𝑔 : 𝐴 → 𝐵 to be the function defined by ∀𝑎 ∈ 𝐴, 𝑓•𝑔 𝑎 =
𝑓 𝑎 •𝑔 𝑎 .
 +, × (“plus”, “times”) are binary operators over R. (Normal addition and
multiplication).
 Therefore, we can also add and multiply functions 𝑓, 𝑔. 𝑅 → 𝑅
 𝑓 + 𝑔 : 𝑅 → 𝑅, where 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔 𝑥
 𝑓 × 𝑔 : 𝑅 → 𝑅, where 𝑓 × 𝑔 𝑥 = 𝑓 𝑥 × 𝑔 𝑥
28 November 2022
Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Constructing Function Operators
 For functions 𝑔: 𝐴 → 𝐵 and 𝑓: 𝐵 → 𝐶, there is a special operator called
“Compose, ∘ ”. [Composition Function]
 It composes (creates) a new function out of 𝑓, 𝑔 by applying 𝑓 to the results
of 𝑔.
 𝑓 ∘ 𝑔 : 𝐴 → 𝐶, where 𝑓 ∘ 𝑔 𝑎 = 𝑓(𝑔 𝑎 ).
 Note 𝑔(𝑎) ∈ 𝐵, So 𝑓(𝑔 𝑎 ) is defined and ∈ 𝐶.
 Note that “∘” (like Cartesian ×, but unlike +, ∧, ∪) is non-commuting.
(Generally, 𝑓 ∘ 𝑔 ≠ 𝑔 ∘ 𝑓
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Example:
Let 𝑓 𝑥 = 2𝑥 + 3 and 𝑔 𝑥 = 3𝑥 + 2, then:
 𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥
= 2 3𝑥 + 2 + 3 = 6𝑥 + 4 + 3 = 6𝑥 + 7
 𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥
= 3 2𝑥 + 3 + 2 = 6𝑥 + 9 + 2 = 6𝑥 + 11
One-to-One Functions:
 A function is one-to-one (1-1), or injective, or an injection iff every element of
its range has only one pre-image.
 Formally, given 𝑓: 𝐴 → 𝐵, 𝑥 is injective” ∶≡ ¬∃𝑥, 𝑦 𝑥 ≠ 𝑦 ∧ "𝑓(𝑥) = 𝑓(𝑦)"
 I mean, there no exist two pre-images give one image.
28 November 2022
Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
 Only one element of the domain is mapped to any given one element of the
range.
 Domain and range have the same cardinality.
What about the codomain?!! [it could be greater or equal to the range)
 Each element of the domain is injected into a different element of the range.
28 November 2022
Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Example:
“one – to – one ”
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Not “one – to – one ”
Not “one – to – one ”
Two pre-images give
one image
One input gives two
outputs (not a function)
Dr. M. Alkrunz
EEE247: Discrete Mathematics
12
Functions,,,
Sufficient Condition for (1-1) Functions:
 For functions 𝑓 over numbers,
 𝑓 is strictly (or monotonically) increasing iff 𝑥 > 𝑦 → 𝑓 𝑥 > 𝑓(𝑦) for all 𝑥, 𝑦
in the domain.
 𝑓 is strictly (or monotonically) decreasing iff 𝑥 > 𝑦 → 𝑓 𝑥 < 𝑓(𝑦) for all 𝑥, 𝑦
in the domain.
Strictly
Increasing
Function
𝒇(𝒙)
𝒇(𝒚)
𝒇(𝒚)
𝒇(𝒙)
𝒚
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Strictly
Decreasing
Function
𝒙
𝒚
Dr. M. Alkrunz
EEE247: Discrete Mathematics
𝒙
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Functions,,,
 If 𝑓 is either strictly increasing or strictly decreasing, then 𝑓 is one to one.
 Example: 𝑥 3
 Converse is not necessarily true. Example: 1Τ𝑥 [at zero we have 2 values]
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Onto Functions:
 A function 𝑓. 𝐴 → 𝐵 is onto or surjective or surjection iff its range is equal to
its codomain.
 ∀ 𝑏 ∈ 𝐵, ∃ 𝑎 ∈ 𝐴: 𝑓 𝑎 = 𝑏
[I mean there exist a value in the domain
produce the range, or all elements in the codomain could be produced from
domain].
 An Onto function maps the set A onto (over, covering) the entirely of the set
B, not just over a piece of it.
 E.g., for domain and codomain “R”, 𝑥 3 is Onto, whereas 𝑥 2 is not [WHY?!!]
 In 𝑥 3 , the range and codomain is the real numbers, R.
 In 𝑥 2 , the range and codomain is not the real numbers, R. Namely, the
codomain is the real numbers while the range is the positive real
numbers.
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Example:
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Bijections:
 A function 𝑓 is one-to-one correspondence, or a bijection, or reversible, or
invertible, iff it is both One-to-one and Onto.
 For bijections 𝑓: 𝐴 → 𝐵, there exists an inverse of 𝑓, written 𝑓 −1 : 𝐴 → 𝐵
which is the unique function such that 𝑓 −1 ∘ 𝑓 = 𝐼 (identity Function)
Identity Function:
 For any domain “A”, the identity function 𝐼: 𝐴 → 𝐴 (written as, 𝐼𝐴 ) is the
unique function such that (∀ 𝑎 ∈ 𝐴: 𝐼 𝑎 = 𝑎)
 Examples: adding to zero, multiplying by one, AND with true, OR with
false, Union with null, Intersection with domain.
28 November 2022
Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
A couple of Key Functions:
 In discrete math, we will frequently use the following functions over real
numbers:
 𝑥 (“floor of x”) is the largest (greatest integer function) integer ≤ 𝑥.
 𝑥 (“ceiling of x”) is the smallest integer ≥ 𝑥.
Visualizing Floor and Ceiling Functions:
 Real numbers “fall to their floor” or “rise to their ceiling”.
 Note that if 𝑥 ∉ 𝑍, then −𝑥 ≠ − 𝑥 and −𝑥 ≠ − 𝑥
 Note that if 𝑥 ∈ 𝑍, then
𝑥 = 𝑥 = 𝑥.
−𝟑
𝟐 1.6 =….,
𝟏 −1.4 =…..,
−𝟏 −1.4 =…..,
−𝟐 −3 = −3 =…..
 Example: 1.6 =….,
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Functions,,,
Plots with Floor/Ceiling:
 Sets that do not include all of their limit points are called “Open Sets”.
 In a plot, we draw a limit point of a curve using an open dot (circle) if the limit
point is not on the curve, and with a closed dot (solid) if it is on the curve.
𝒙
𝒙
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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Exercise,,,
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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*Questions
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EEE247: Discrete Mathematics
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*Thanks
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Dr. M. Alkrunz
EEE247: Discrete Mathematics
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