Discrete Mathematics Answers

20. Determine the truth value of each statement if the domain consists of all real numbers. (4 pts.)

a) ∃𝑥(𝑥3 = −1) b) ∃𝑥(𝑥4 < 𝑥2) c) ∀𝑥((−𝑥)2 = 𝑥2) d) ∀𝑥(2𝑥 > 𝑥)

Let Q(x) be the statement “x + 1 > 2x.” If the domain consists of all integers, what are these truth values? (7 pts.)

a) 𝑄(0) b) 𝑄(−1) c) 𝑄(1)

d) ∃𝑥𝑄(𝑥) e) ∀𝑥𝑄(𝑥) f) ∃𝑥¬𝑄(𝑥) g) ∀𝑥¬𝑄(𝑥)

Let 𝐴={1,2,3,4}. Determine the truth value of each statement:

i. ∀𝑥 ∈𝐴,𝑥+3<6

ii. ∃𝑥,𝑥+3<6

iii. ∃𝑥,2𝑥2 +𝑥 =15

Let 𝐴={1,2,3,4}. Determine the truth value of each statement:

i. ∀𝑥 ∈𝐴,𝑥+3<6

A class has 175 students. The following data shows the number of students taking one or more subjects. Mathematics 100, Physics 70, Chemistry 40; Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. Use a venn diagram to show how students are taking maths alone

Q1 Let R = {(1,4),(2,1),(2,5),(2,4),(4,3),(5,3),(3,2)}. Use warshall’s algorithm to find the matrix of transitive closure where A = {1, 2, 3, 4, 5}

Given that n is a positive integer and in the expansion of (1+ax)ⁿ, the coefficient of x is 200. Find the coefficient of x in the expansion of (1+ax)²ⁿ.

Find ⋃ 𝑨𝒊 ∞ 𝒊=𝟏 and ⋂ 𝑨𝒊 ∞ 𝒊=𝟏 where:

𝑨𝒊 = {𝒊,𝒊 + 𝟏,𝒊 + 𝟐, … } for every positive integer 𝒊.

Find the generating function of recurrence relation a_(n+1) - a_n = 3n ,n<0 where ao=1

Show by giving a proof by contrapositive, that if 3n+2 is odd, then n is odd