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Discrete Mathematics
Higher Nationals
Internal verification of assessment decisions – BTEC (RQF)
INTERNAL VERIFICATION – ASSESSMENT DECISIONS Programme title BTEC Higher National Diploma in Computing Assessor Internal Verifier Unit(s) Unit 18 : Discrete Mathematics Assignment title Discrete mathematics in software engineering concepts Student’s name S,DILAKSON List which assessment criteria the Assessor has awarded. Pass Merit Distinction INTERNAL VERIFIER CHECKLIST Do the assessment criteria awarded match those shown in the assignment brief? Y/N Is the Pass/Merit/Distinction grade awarded justified by the assessor’s comments on the student work? Y/N Has the work been assessed accurately? Y/N Is the feedback to the student: Give details:
- Constructive?
- Linked to relevant assessment criteria?
- Identifying opportunities for improved performance?
- Agreeing actions? Y/N Y/N Y/N Y/N Does the assessment decision need amending? Y/N Assessor signature Date Internal Verifier signature Date Programme Leader signature (if required) Date 1
Confirm action completed Remedial action taken Give details: Assessor signature Date Internal Verifier signature Date Programme Leader signature (if required) Date 2
Pearson Higher Nationals in Computing Unit 18: Discrete Mathematics 4
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Assignment Brief
Student Name /ID Number S.DILAKSON 15077 Unit Number and Title Unit 18: Discrete Mathematics Academic Year Unit Tutor Assignment Title Discrete mathematics in Computing Issue Date Submission Date IV Name & Date Submission Format: This assignment should be submitted at the end of the module, on the week stated at the front of this brief. The assignment can either be word-processed or completed in legible handwriting. If the tasks are completed over multiple pages, ensure that your name and student number are present on each page. Unit Learning Outcomes: LO1 Examine set theory and functions applicable to software engineering LO2 Analyze mathematical structures of objects using graph theory LO3 Investigate solutions to problem situations using the application of Boolean algebra LO4 Explore applicable concepts within abstract algebra. 7
Contents
Activity 01
Part 01
1. A ∪ B^?
2. n ( B )? 3. n ( A ∪ B ∪C )=? n ( A ) = 33 n ( B )= 36 n ( C )= 28 11 n( A B ) = n(A) + n(B) - n( A B )
n( A B ) = 72 + 28 - 13
n( A B ) = 87
n( A B ) =
n( A B ) =
n( A B ) =
n( A B ) = n( A B ) + n( B A ) + n( A B )
110 = 45 + n( B A ) + 15 110 = 60 + n( B A ) n( B A ) = 50
n(B) = n( B A ) + n( A B )
n(B) = 50 + 15 n(B) = 65 Figure 1 Venn diagram
Part 02
1. Multiple prime factors i. 160 = {2, 2, 2, 2, 2, 5} Multiplicity of 2 = 5 Multiplicity of 5 = 1 ii. 120 = 2, 2, 2, 2, 3, 5} Multiplicity of 2 = 3 Multiplicity of 3 = 1 Multiplicity of 5 = 5= iii. 250 = {2, 5, 5, 5} Multiplicity of 2 = 2 Multiplicity of 5 = 3 2. Cardinalities of each multiset i. Cardinality of multi set = 5 + 1 = 6 ii. Cardinality of multi set = 3 + 1 +1 = 5 iii. Cardinality of multi set = 2 + 3 = 5
Part 03
1. Determine whether the following functions are invertible or not. If it is
invertible, then find the rule of the inverse (^ f
− 1
( x ) )
i. f : ℜ→ℜ+^ ii. f : ℜ+→ ℜ+ f ( x )= x^2 f ( x )=
x iii. f : ℜ+→ ℜ+^ iv. f :[ − π 2
π 2 ]
→[− 1 , 1 ]
f ( x )= x^2 f ( x )=sin x
v. f :[ 0 , π ] →[ − 2 , 2 ]
f ( x )= 2 cos x 13
N = {1, 2, 3, 4, 5}
Z = {…. -3, -2, -1, 0, 1, 2, 3, 4, 5….}
Z+^ = {1, 2, 3, 4, 5….}
Z-^ = {…. -3, -2, -1,}
I. f^ (^ x^ )= x
2 f ( x )= 2x f (^) (-3) = 2(-3) = (-6) → f (^) 2 * (-6) = (-12) f (^) (-2) = 2(-2) = (-4) → f (^) 2 * (-4) = (-8) f (-1) = 2(-1) = (-2) → f (^) 2 * (-2) = (-4) f (0) = 2(0) = (0) → f (^) 2 * (0) = (0) f (^) (1) = 2(1) = (2) → f (^) 2 * (2) = (4) f (^) (2) = 2(2) = (4) → f (^) 2 * (4) = (8) f (3) = 2*(3) = (6) → f (^) 2 * (6) = (12) II. f ( x )= 1 x f ( 1 )= 1 1 = 1 14
2 4 f ( 2 )= 22 = 4
4 16 f ( 4 )= 42 = 16
5 25 f ( 5 )= 52 = 25
25 625 f ( 25 )= 252 = 625
0.1 0.01 f ( 0. 1 )=( 0. 1 )^2 = 0. 01
f ( 0. 01 )=( 0. 01 )
2
Table 1 Process X 1 , X 2 be two different numbers from domain f ( x 1 )= x (^12) →① f ( x 2 )= x (^22) →② ① = ② X 12 = X 22 Take the square root X 1 = X 2 Then 1 – 1 function Inverse exit.
f ( x )=
x )
2 f ( x )= y y = x^2 Take the square root √ y =√ x 2 √ y = x x = y & y = x √ x = y (Inverse)
f
− 1 ( (^) x ) (^) = √ x 16
I.
f ( x )=sin x
f :[− π 2 , π 2 ] →[− 1 , 1 ]
f (
)=sin(
= - sin
f (
)=sin (
f (
)=sin(
= -sin
f (
)=sin (
=sin
√^3
Sin 0 = 0 onto function X 1 , x 2 be two different numbers f (x 1 ) = sin (x 1 ) 01 f (^) (x 2 ) = sin (x 2 ) 02 01 = 02 Sin (x 1 ) = sin (x 2 ) [ − π 2 , π 2 ] f (^) (x) = sin (x) y = f (x) y = sin (x) 17 Sin (-x) = - Sin (x)
iv. f ( x )= 2cosx f ( x 1 )= 2cos(x1) 01 f ( x 2 )= 2cosx(x2) 02 01 = 02 X 1 = X 2 So 1 – 1 function Y = 2Cosx cos − 1 ( y 2 ) = x Y = cos − 1 ( x 2 ) f-1^ (x) = cos − 1 ( x 2 ) 2.
f ( x )=
( x − 32 )
f (^) (98.6) =
97^0 C
f (x) = y y =
(x – 32) 9 y 5
- 32 = x y → x , x → y Y= 9 x 5
19
f-1^ (x) = 9 x 5
Part 04
1. A = b ↔ A ⊆ B ∧ B⊆ A Definition : Two sets are equal if they contain the same elements. I.e., sets A and B are equal if ∀x[x ∈ A ↔ x ∈ B]. Notation : A = B. Recall : Sets are unordered and we do not distinguish between repeated elements. So: {1, 1, 1} = {1}, and {a, b, c} = {b, a, c}. Definition : A set A is a subset of set B, denoted A ⊆ B, if every element x of A is also an element of B. That is, A ⊆ B if ∀x(x ∈ A → x ∈ B). Example : Z ⊆ R. {1, 2} ⊆ {1, 2, 3, 4} Notation : If set A is not a subset of B, we write A⊆ B. Example : {1, 2} 6⊆ {1, 3} 2. De Morgan’s Law by the mathematical induction 20
