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For any natural number n, prove the validity of given series by mathematical induction:
2(√(n+1)-1)<1+(1/√2)+⋯..+(1/√n)<2√n?
Make your own example with at least four restrictions and one conclusion.
Description:
p∨q∨r∨s→t=(p→t)∧(q→t)∧(r→t)∧(s→t)?
In a school, 863+100 students have access to three software packages A, B and C. 862 did not use any software. 861 used only packages A. 860 used only packages B. 859 used only packages C. 858 used all three packages. 857 used both A and B. (1) Draw a Venn diagram with all sets enumerated as for as possible. Label the two subsets which cannot be enumerated as x and y in any order. (2) If twice as many students used package B as package A, write down a pair Of simultaneous equations in x and y. (3) Solve the equations to find x and y. (4) How many students used package C?
Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is 863
In a school, n+100 students have access to three software packages A, B andC n-1 did not use any software , n-2 used only packagesA n-3 used only packages B , n-4 used only packages C
n-5 used all three packages , n-6 used both A and B.
where n is 158
a) Draw a Venn diagram with all sets enumerated as for as possible.Label the two subsets which cannot be enumerated as x and y in any order.
b) If twice as many students used package B as package A,write down a pair of simultaneous equations in x and y. C)Solve the equations to find x and y. D) How many students used package C?
Q.No.5. [2+1+3]
a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is even digit of your arid number e.g 19-arid-234 take n=2
Question 5: By using the rules of logical equivalences, show the propositions are logically equivalent:
a) Determine whether (p → (q → r)) → (p ˄ q) → r) is Tautology.
b) (p ∧ q) ∧ [(q ∧ ¬r) ∨ (p ∧ r)] and ¬(p → ¬q).
c) [(p v q) /\ (p → r) /\ (q → r)] →r is Tautology.
: Let p and q be the propositions.
p: “I bought a lottery ticket.”
q: “I won the million dollar jackpot on Friday.”
Express each of these propositions as an English sentence.
a) ￢p b) ￢p →￢q
c) p ↔q d) ￢p ∨(p ˄ q)
Write Inverse, Converse and Contrapositive, also apply implication law on the following statements:
a)If it snows today. I will ski tomorrow.b)I come to class whenever there is going to be a quiz.c)I go to the beach whenever it is a sunny summer day.d) When I stay late, it is necessary that I sleep until noon.
