Solution to PROBLEM SOLVING. A. SET. Let A, B and C are sets and U be universal … - Sikademy
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Archangel Macsika

PROBLEM SOLVING. A. SET. Let A, B and C are sets and U be universal set. U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e} A = {-1, 1, 2, 4} B = {0, 2, 4, 6} C = {b, c, d} Find for the following. Show complete solutions. 1. 𝐡 βˆͺ 𝐢 2. 𝐴 βˆ’ 𝐡 π‘₯ 𝐢 3. π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝑠𝑒𝑑 π‘œπ‘“ 𝐢 4. |𝑃(𝐡)| B. SEQUENCES. Consider the sequence {Sn} defined by Sn = 2n – 5, where 𝒏 β‰₯ βˆ’πŸ. Find for: 1. βˆ‘1𝑖=βˆ’1 𝑆𝑖 (5 pts) C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x β‰₯ y. Find for: 1. Elements of R 2. Domain and Range of R 3. Draw the digraph 4. Identify the properties of R

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Solution:

(A):

U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e}

A = {-1, 1, 2, 4}

B = {0, 2, 4, 6}

C = {b, c, d}

1. 𝐡 βˆͺ 𝐢 = {0, 2, 4, 6, b, c, d}

2. 𝐡 \times 𝐢=\{(0,b),(0,c),(0,d),(2,b),(2,c),(2,d),(4,b),(4,c),(4,d),(6,b),(6,c),(6,d)\}

Now,Β A-B\times C=\{-1, 1, 2, 4\}

3. π‘ƒπ‘œπ‘€π‘’π‘Ÿ 𝑠𝑒𝑑 π‘œπ‘“ 𝐢=\{\phi,\{b\}, \{c\},\{d\},\{b,c\},\{c,d\},\{b,d\},\{b,c,d\}\}

4. |𝑃(𝐡)|=2^nΒ , where n is the number of elements in set B.

|P(B)|=2^4=16

(B):

S_n=2n-5,n\ge-1

\Sigma_{-1}^1 S_i=S_{-1}+S_0+S_1 \\=2(-1)-5+2(0)-5+2(1)-5 \\=-2+0+2-15 \\=-15

(C):

Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if xΒ β‰₯Β y.

1.

R=\{(-3,-3),(-2,-2),(-1,-1),(0,0),(1,1),(-2,-3),(-1,-3),\\(0,-3),(1,-3),(-1,-2),(0,-2),(1,-2),(0,-1),(1,-1)(1,0)\}

2. Domain of R=\{-3,-2,-1,0,1\}

And range of R=\{-3,-2,-1,0,1\}

3. Digraph of R:



4.

Reflexive:

It is clearly reflexive asΒ (a,a)\in R, \forall a\in X

Symmetric:

It is clearly not symmetric asΒ a\ge bΒ butΒ b\ge aΒ is not true,Β \forall a,b \in X

Moreover,Β (0,-2)\in RΒ butΒ (-2,0)\not\in R

Transitive:

a\ge b, b\ge c \Rightarrow a\ge cΒ which is trueΒ \forall a,b,c \in X

Hence, it is transitive.


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Question ID: mtid-5-stid-8-sqid-3129-qpid-1828