Answer to Question #183060 in Discrete Mathematics for Ae dion

Question

. 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. (3 pts each)

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. ∑𝑆𝑖1𝑖=−1

2. ∏𝑆𝑖4𝑖=2

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

Find for:

1. Elements of R (3 pts)

2. Domain and Range of R (2 pts)

3. Draw the digraph (3 pts)

4. Identify the properties of R (2pts)

Answer 1

A.

1  B$\cup$ C={0,2,4,6,b,c,d}

2.$A-B \times C$ ={-1,1} x {b,c,d}={(-1,b),(-1,c),(-1,d),(1,b),(1,c),(1,d)}

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

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

B. 1.$\sum_{i=-1}^{i=1}S_i=\sum _{i=-1}^{i=1}2i+5=2(-1)+5+2(0)+5+2(1)+5=15$

2.$\sum_{i=2}^{i=4}S_i=\sum _{i=2}^{i=4}(2i+5)=2(2)+5+2(3)+5+2(4)+5=4+6+8+15=33$

C.X={-3,-2,-1,0,1}

Relation R is defined as-

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

1.Elements of R are (-2,-3),(-1,-3),(-1,-2),(0,-3),(0,-2),(0,-1),(1,-3),(1,-2),(1,-1),(1,0)

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

Range of R ={-3,-2,-1,0}

3.Diagraph is-

4.Properties of R-

(i) R is the subset of the cartesian product form from the given set.

(ii) R can be reflexive,transitive and symmetric in nature.