Discrete math questions

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oruv_1306

Mathematics

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1) Nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. In how many ways can this be done if Ben is to be in the picture?

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1) Nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. In how many ways can this be done if Ben is to be in the picture? 2) Find the number of subsets of S={1,2,3,...,10}thatcontainthenumber5. 3) See Page 169 of the Textbook Ex. 33 d. 4) If 1+1=2, then 2+2=5. True or False? 5) Prove that if n is a positive integer, then n is odd if and only if 5n+6 is odd. 6) See Page 184 of the textbook Ex. 27 c. 7) Determine whether the functions f(x)= 2x+1 is a bijection from R to R. 8) Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4 9) Consider the relation {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} on the set{1,2,3,4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. 10)Draw the directed graph of the reflexive closure of the relations with the following directed graph. Extra Points: 11) Prove that 12 +32 +52 +···+(2n+1)2 = (n+1)(2n+1)(2n+3)/3 whenever n is an nonnegative integer. 12) How many 5-element DNA sequences do not contain C?
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Explanation & Answer

The solutions are ready. Please ask if something is unclear.I am still ready to solve 3, 6, 10 when you'll give me them.

1. It is not clear whether the order in a row is significant.
9
9
If not, there are ( ) = ( ) combinations of 5 people. Ben is to be in the picture means that
5
4
8
8
those ( ) = ( ) combinations without Ben are not suitable. This way we obtain
5

3

9∙8∙7∙6 8∙7∙6
9
8
( )−( )=

= 9 ∙ 2 ∙ 7 − 8 ∙ 7 = 10 ∙ 7 = 𝟕𝟎.
4
3
4∙3∙2∙1 3∙2∙1

If the order is significant, then there are 𝑃(9,5) − 𝑃(8,5) =
= 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 − 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 = 8 ∙ 7 ∙ 6 ∙ 5 ∙ 5 = 𝟖𝟒𝟎𝟎.

2. Subsets are always considered without order. Therefore, we fix the element 5 and unite
it with any subset of {1, 2, 3, 4, 6, 7, 8, 9, 10}. There are 9 elements in this set, so it has
29 = 𝟓𝟏𝟐 subsets, which is the answer.

4. False.
We have an implication 𝐴 ⇒ 𝐵 here. There is the only option where it considered to be
false, namely 𝐴 is true and 𝐵 is false.
We take correct sentence 1 + 1 = 2 (with the truth value of true) and draw a false
conclusion 2 + 2 = 5, so this implication is false.

5. 𝑛 is odd if and only if 5𝑛 + 6 is odd (it is correct for any integer, not for positive only).
If: let 𝑛 is odd, then 5 ∙ 𝑛 is odd (odd by odd) and 5𝑛 + 6 is odd, too (odd + even).
Only if: let 5𝑛 + 6 is odd, then 5𝑛 is odd (odd - even). When 5𝑛 is odd, 𝑛 cannot be even,
so it is odd, Q.E.D.

7. 𝑓: ℝ → ℝ, 𝑓(𝑥) = 2𝑥 + 1.
Yes, it is a bij...


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