Answer to Question #183689 in Discrete Mathematics for Angelie Suarez

Question #183689

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. 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)

2. ∏ 𝑆𝑖 4𝑖=2 (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 (3 pts)

2. Domain and Range of R (2 pts)

3. Draw the digraph (3 pts)

4. Identify the properties of R (2pts)


1
Expert's answer
2021-05-07T09:02:10-0400

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. 𝐡 βˆͺ 𝐢 = {0, 2, 4, 6, b, c, d}

2. "\ud835\udc35 \\times \ud835\udc36=\\{(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, "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^n" , 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\n\\\\=2(-1)-5+2(0)-5+2(1)-5\n\\\\=-2+0+2-15\n\\\\=-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 as "(a,a)\\in R, \\forall a\\in X"

Symmetric:

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

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

Transitive:

"a\\ge b, b\\ge c \\Rightarrow a\\ge c" which is true "\\forall a,b,c \\in X"

Hence, it is transitive.


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