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Selasi Justice

Selasi J.

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Questions asked

INSTANT ANSWER

6. (a) Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x)=\frac{x^{2}-2}{x} \). (i) Find the Domain and Range of the function \( f \). (ii) Show that the function defined above is injective (one-to-one). (iii) Using the range obatined in (i), find the inverse \( f^{-1} \) of the function. (b) For each of the following determine the value(s) of the integer \( n>1 \) for which the given congruence is true (i) \( 28 \equiv 6(\bmod n) \). (ii) \( 68 \equiv 37(\bmod n) \).

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INSTANT ANSWER

5. (a) Prove that following statements, where \( A, B \) and \( C \) are sets. (i) \( \left[(A \cap B) \cup\left(A^{\prime} \cap C\right)\right]^{\prime}=\left(A \cap B^{\prime}\right) \cup\left(A^{\prime} \cap C^{\prime}\right) \). (ii) \( (A \backslash B) \cap(A \backslash C)=A \backslash(B \cup C) \). (iii) \( (A \oplus B)=(B \oplus A) \). (b) Show that compound statement \( [(p \Rightarrow q) \Rightarrow r] \) is logically equivalent to the compound statement \( [(p \vee r) \wedge(q \Rightarrow r)] \).

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INSTANT ANSWER

Determine \( \operatorname{gcd}(a, b) \) and express it as a linear combination of \( a \) and \( b \), where \( a=23 \) \( b=1820 \). Obtain the congruence classes of integer modulo 5 that is \( a \equiv b(\bmod 5), a, b \in \mathbb{Z} \).

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INSTANT ANSWER

2. Use the principle of mathematical induction to prove the following statements (a) \( \sum_{i=1}^{n}(2 i-1)^{2}=\frac{n(2 n-1)(2 n=1)}{3} \). (b) \( 2^{2 n+1}+1 \) is divisible by 3.6 (c) If \( A_{1}, A_{2}, \ldots, A_{n} \) are subsets of the universal set \( U \), then \( \left(\bigcap_{i=1}^{n} A_{i}\right)^{\prime}=\bigcup_{i=1}^{n} A_{i}^{\prime} \), where \( A_{i}^{\prime} \) denotes the complement of the set \( A_{i} \).

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INSTANT ANSWER

Prove that for every \( x, y \in \mathbb{R} \), if \( x+y \geq 100 \) then \( x \geq 50 \) or \( y \geq 50 \). Let \( x, y \in Z \). Prove that if \( a \) divides \( b \) and \( a \) divides \( c \) then \( a \) divides \( b x+c y \). Let \( a, b, c, d \in \mathbb{Z}^{+} \). Prove that if \( a \) divides \( b \) and \( c \) divides \( d \) then \( a c \) divides \( b d \). Let \( a, b \) and \( c \) be positive integers. Prove that if \( a \) divides \( b c \) then \( a \) divides \( b \) or \( a \) divides \( c \).

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INSTANT ANSWER

(a)Letf:R→Rbeafunctiondefinedbyf(x)=x2−2. x (i) Find the Domain and Range of the function f. (ii) Show that the function defined above is injective (one-to-one). (iii) Using the range obatined in (i), find the inverse f−1 of the function. (b) For each of the following determine the value(s) of the integer n > 1 for which the given congruence is true (i) 28 ≡ 6(mod n). (ii) 68 ≡ 37(mod n).

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INSTANT ANSWER

a) Prove that following statements, where A, B and C are sets. (i)[(A∩B)∪(A′ ∩C)]′ =(A∩B′)∪(A′ ∩C′). (ii) (A\B) ∩ (A\C) = A\(B ∪ C). (iii) (A ⊕ B) = (B ⊕ A). (b) Show that compound statement [(p ⇒ q) ⇒ r] is logically equivalent to the compound statement [(p ∨ r) ∧ (q ⇒ r)].

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INSTANT ANSWER

(a) For each of these relations on the set {1, 2, 3, 4}. Decide whether it is reflexive, symmetric and transitive. Justify your claims. R1 ={(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} R2 ={(1,2),(1,2),(2,1),(2,2),(3,3),(4,4)} R3 ={(2,4),(4,2)} R4 ={(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)} (b) Let R be the relation on the set of ordered pairs of positive integers such that (a, b)R(c, d) if and only if a+d = b+c. Show that R is an equivalence relation on the positive integers

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INSTANT ANSWER

a) Determine gcd(a, b) and express it as a linear combination of a and b, where a = 231 and b = 1820. (b) Obtain the congruence classes of integer modulo 5 that is a ≡ b(mod 5), a, b ∈ Z

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INSTANT ANSWER

Use the principle of mathematical induction to prove the following statements (a) Pn (2i − 1)2 = n(2n−1)(2n=1) . i=1 3 (b) 22n+1 + 1 is divisible by 3.6 (c) If A1, A2, ..., An are subsets of the universal set U, then (Tni=1 Ai)′ = Sni=1 A′i, where A′i denotes the complement of the set Ai.

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