In November 2024
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Let f : R → R be defined by f(x) = (x3 + 1)/2
a. Prove that f is bijective
b. Determine f-1 (x) and f o f o f-1
For sets A = {-3, -2,…,3} and B = {0, 1,…,10} B’ = {0, 1, 4, 5, 8, 9} and C = {1, 2,…,10}, let f : A → B and g : B’ → C be functions defined by f(n) = n2 for all n ∈ A and g(n) = n + 1 for all n ∈ B’.
a. Show that the composition g o f : A → C is defined
b. For n"\\in" A, determine (g o f)(n)
Show that (~𝒑∨𝒒)∧(𝒑∧~𝒒) is a contradiction
2. How many 4 letter words can be created, if repetitions are allowed?
3. How many three digit numbers can be formed?
4. How many words (of any number of letters) can be formed from GAME?
5. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46723819?
6. How many numbers between 999 and 9999 are divisible by 5 and have no repeated digits?
7. How many ways can you order the letters in KEYBOARD if K and Y must always be kept together?
8. How many ways can 4 rock, 5 pop, and 6 classical albums be ordered if all albums of the same genre must be kept together?
9. In how many ways can the letters from the word EDITOR be arranged if vowels and consonants alternate positions?
10. If 8 boys and 2 girls must stand in line for a picture, how many line-up’s will have the
Let C(x) be the statement “x can play chess,” let D(x) be the statement “x can play table tennis. Express each of following statements in terms of C(x), D(x), quantifiers, and logical connectives.
Let the domain consist of all students in your class.
i) A student in your team can play chess but do not know how to play table tennis.
ii) All students in your team can play chess.
iii) Some students in your team can not play table tennis.
iv) No student in your team can play table tennis or chess.
2. How many 4 letter words can be created, if repetitions are allowed?
3. How many three digit numbers can be formed?
4. How many words (of any number of letters) can be formed from GAME?
5. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46723819?
6. How many numbers between 999 and 9999 are divisible by 5 and have no repeated digits?
7. How many ways can you order the letters in KEYBOARD if K and Y must always be kept together?
8. How many ways can 4 rock, 5 pop, and 6 classical albums be ordered if all albums of the same genre must be kept together?
9. In how many ways can the letters from the word EDITOR be arranged if vowels and consonants alternate positions?
10. If 8 boys and 2 girls must stand in line for a picture, how many line-up’s will have the
2. How many 4 letter words can be created, if repetitions are allowed?
3. How many three digit numbers can be formed?
4. How many words (of any number of letters) can be formed from GAME?
5. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46723819?
6. How many numbers between 999 and 9999 are divisible by 5 and have no repeated digits?
7. How many ways can you order the letters in KEYBOARD if K and Y must always be kept together?
8. How many ways can 4 rock, 5 pop, and 6 classical albums be ordered if all albums of the same genre must be kept together?
9. In how many ways can the letters from the word EDITOR be arranged if vowels and consonants alternate positions?
10. If 8 boys and 2 girls must stand in line for a picture, how many line-up’s will have the girls separated from each other?
Express the negation of each of these statements in terms
of quantifiers without using the negation symbol.
a) ∀x(x > 1)
b) ∀x(x ≤ 2)
c) ∃x(x ≥ 4)
d) ∃x(x < 0)
e) ∀x((x < −1) ∨ (x > 2))
f ) ∃x((x < 4) ∨ (x > 7))
Show that (~𝒑∨𝒒)∧(𝒑∧~𝒒) is a contradiction.
Construct a graph G with 6 vertices {v1, v2, v3, v4, v5, v6} and six edges {e1, e2, e3, e4, e5,
e6} such that
i. e2 is a loop at v2
ii. v2 and v5 are end point of e5
iii. v3 is adjacent to v2
iv. v4 is isolated
v. e3 is parallel to e5
vi. e4 is incident of v1 and v6
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#339914 on Dec 2023