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. ∑1𝑖=−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 ≥ 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)
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. "\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. 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\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 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.
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