Question

III. 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

2. ∏ 𝑆𝑖 4𝑖=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

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. $𝐵 \times 𝐶=\{(0,b),(0,c),(0,d),(2,b),(2,c),(2,d),(4,b),(4,c),(4,d),(6,b),(6,c),(6,d)\}$

Now, $A-B\times C=\{-1, 1, 2, 4\}$

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

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

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

(B):

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

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

$\Pi_2^4S_i=S_2\times S_3 \times S_4=(4-5)(6-5)(8-5)=(-1)(1)(3)=-3$

(C):

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

1.

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

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.