Discrete Mathematics Answers

Using propositional-formula and logical connectives, convert these sentences into symbolic form. 

1. No doctors are enthusiastic;  

You are enthusiastic.  

Therefore, you are not a doctor  

1. Dictionaries are useful;  

Useful books are valuable.  

Therefore, dictionaries are valuable.   

1. Some healthy people are fat;  

No unhealthy people are strong.  

Therefore, some fat people are not strong.  

1. All, who are anxious to learn, work hard;  

Some of these boys work hard.  

Therefore, some of these boys are anxious to learn.   

1. All lions are fierce;  

Some lions do not drink coffee.  

Therefore, some creatures that drink coffee are not fierce.  

1.  No misers are generous;  

Some old men are ungenerous.  

  Therefore, some old men are misers.  

1. All young lambs jump;  

No young animals are healthy, unless they jump. 

Therefore, all young lambs are healthy.  

1. No professors are ignorant.  

All ignorant people are vain.  

Therefore, no professors are vain.   

1. All bees are unfriendly;  

No butterflies are unfriendly.  

Therefore, butterflies are not bees. 

Show that among 2000 student in a school, at least six were born on the same day of the leap year

Make a truth table for the given expression.

12. (~p∧q) ∨ (p∧~q)

13. (p∧~q) ∨ r

14. ~[(p∧~q) ∨ ~p]

15. (p∨q) ∧ ~(~q∧r)

16. (~p∧q) ∨ (~p∨q) 17. ~[(p∨q) ∧ ~q]

The Mathclub, VIT-AP wants to conduct a group event for its members. So the club president has to fix the group size the event. When he tries to fix the size to be 5 members in each group, 4 members are left; when he tries to fix the size to be 6 members in each group, 5 members are left; When he fixes the size to be 7 members in each group, 6 members are left. What is the smallest number of members that the club has?

Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

36. Show that the propositions pi, p2, p3, and p4 can be shown to be equivalent by showing that p1 ↔ p4, P2 ↔ P3, and p1 ↔ p3.

the number of combinations of five objects taken two at a time is?

Let ρ be a relation on a set A. Define ρ −1 = { | ∈ ρ}. Also for two relations ρ, σ on A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that ∈ ρ and ∈ σ. Prove the following assertions (a) ρ is both symmetric and antisymmetric if and only if ρ ⊆ { | a ∈ A}. (b) ρ is transitive if and only if ρ ◦ ρ = ρ. 

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading every exam may be checked by any number of graders (but always at least by one grader). This means that summed all together the graders may make up to k ∗ n exam checks. To avoid this it is required that for each pair of graders there is at most 1 exam that they have both checked. Prove that this rule creates a much better bound of at most ((k +n) 3/2 + (k +n))/2 exam checks. Hint: Consider modeling the grading work as a graph. 

Find the sum of product expansion of the Boolean function f(x,y,z) = (x+z)y