Question #183509

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. 𝐵 ∪ 𝐶

2. 𝐴 − 𝐵 𝑥 𝐶

3. 𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡 𝑜𝑓 𝐶

4. |𝑃(𝐵)|

B. SEQUENCES. Consider the sequence {Sn} defined by Sn = 2n – 5, where 𝒏 ≥ −𝟏.

Find for:

1. ∑1𝑖=−1 𝑆𝑖 (5 pts)

C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) R if x y.

Find for:

1. Elements of R

2. Domain and Range of R

3. Draw the digraph

4. Identify the properties of R


1
Expert's answer
2021-04-27T01:44:59-0400

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. 𝐵 ∪ 𝐶 = {0, 2, 4, 6, b, c, d}

2. 𝐵×𝐶={(0,b),(0,c),(0,d),(2,b),(2,c),(2,d),(4,b),(4,c),(4,d),(6,b),(6,c),(6,d)}𝐵 \times 𝐶=\{(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, AB×C={1,1,2,4}A-B\times C=\{-1, 1, 2, 4\}

3. 𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡 𝑜𝑓 𝐶={ϕ,{b},{c},{d},{b,c},{c,d},{b,d},{b,c,d}}=\{\phi,\{b\}, \{c\},\{d\},\{b,c\},\{c,d\},\{b,d\},\{b,c,d\}\}

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

P(B)=24=16|P(B)|=2^4=16

(B):

Sn=2n5,n1S_n=2n-5,n\ge-1

Σ11Si=S1+S0+S1=2(1)5+2(0)5+2(1)5=2+0+215=15\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 x ≥ 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)}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}=\{-3,-2,-1,0,1\}

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

3. Digraph of R:



4.

Reflexive:

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

Symmetric:

It is clearly not symmetric as aba\ge b but bab\ge a is not true, a,bX\forall a,b \in X

Moreover, (0,2)R(0,-2)\in R but (2,0)∉R(-2,0)\not\in R

Transitive:

ab,bcaca\ge b, b\ge c \Rightarrow a\ge c which is true a,b,cX\forall a,b,c \in X

Hence, it is transitive.


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