Discrete Mathematics Answers

Let R and S be relations on X. Determine whether each statement is true or false. If the statement is true, prove it; otherwise, give a counterexample.

1.     If R is transitive, then R⁻¹ is transitive.

2.     If R and S are reflexive, then R ◦ S is reflexive

3.     If R and S are symmetric, then R ∩ S is symmetric.

4.     If R and S are antisymmetric, then R ∪ S is antisymmetric.

5.     If R is antisymmetric, then R⁻¹ is antisymmetric.

Produce a truth table for given Boolean expression (A+B'+C)(A+B+C)(A'+B+C')

In a school 844 students have access to three software packages A, B, C.

Where 743 didn’t use any software, 740 used only package C, 742 used only package

A,741 used package B, 739 used all three packages, 738 used both A and C

a) Draw a Venn diagram with all sets enumerated as for as possible.

b) If twice as many students used package A as Package C, write down a pair

of simultaneous equations in x and y.

c) Solve these equation to find x and y.

d) How many students used package B.

(b) For any natural number n, prove the validity of given series by mathematical induction:

In a school, n+100 students have access to three software packages A, B and C.n-1 did not use any software  ,  n-2 used only packages A  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 your arid number i.e. 19-arid-234 take n=234 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?

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?

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?

 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.