In June 2024
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Prove the 3√7 is irrational
Show that if an integer n is not divisible by 3, then n2 – 1 must be a multiple of 3
Find the domain and range of these functions.
c) the function that assigns to a bit string the number of
times the block 11 appears
d) the function that assigns to a bit string the numerical
position of the first 1 in the string and that assigns the
value 0 to a bit string consisting of all 0s
1.Draw the Hasse diagram representing the partial ordering {(a,b)|a divides b} on {1,2,3,4,6,8,12}
2.Draw the Hasse diagram representing the partial ordering {(a,b)| a divides b} on {1,2,3,4,5,6,10,12,15,20,30,60}.[These are the divisors of 60 which form the basis of the ancient Babylon Ian base- 60 numeral system].
4.Draw the Hasse diagram for the "less than or equal to" relation on {0,2,5,10,11,15}.
5.Draw the Hasse diagram for divisibility on the set
a.{1,2,3,4,5,6}
b.{3,5,7,11,13,16,17}
c.{2,3,5,10,11,15,25}
d.{1,3,9,27,81,243}
8) How many positive integers less than 1000 have at least one decimal digit equal to 9?
solve the following recurrence relations.
a) T(n)=T(n-1)+1 , for n≥ 2 and T(1)=1.
b) "a_{n+1}-2a_n=2n," for n≥1 and T(1)=2.
Three sets have 5, 10, and 15 elements, respectively. How many elements can their union and their intersection have?
Prove that for any three sets 𝐴,𝐵, 𝐶, Venn diagram
((𝐴 \ 𝐵) ∪ (𝐵 \ 𝐴)) ∩ 𝐶 = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) \ (𝐴 ∩ 𝐵 ∩ 𝐶)
We form the intersection of two sets. We know that one of them has n elements and the other has m elements. What can we infer about the cardinality of their intersection?
Let P, Q and R be propositions defined as follows:
P: I am thirsty
Q: My glass is empty
R: It's 3 O'clock
Write each of the following propositions as logical expressions involving P.Q and R
If I am not thirsty , then my glass is not empty. NB: Use "=>" and "~" to represent conditional and negation connectives.
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