Math Answers

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)

Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r

Let ( u1,u2,...un) be an orthogonal basis for a subspace W of R^n and let T:R^n-->R^n be defined by T(x)=proj W(x). Show that T is linear transformation.

Show that T(x1,x2, x3,x4)= 3x1 -7x2+5x4 is liner transformation by finding the matrix for transformation. Then find a basis for the null space of the transformation

Let T:R^n--> R^m be a linear transformation and let ( v1,v2,....v3) be a linearly dependent set. Show that the set ( T (v1),T(v2),....T(vn))is also necessarily linearly dependent.

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p)

Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) \iff p \lor \lnot q \lor \lnot r

find the sum and product of eigenvalues of the matrix

[1 2 3

-1 2 1

1 1 1 ]

Mr. Vijay, a retired government servant, is considering investing his money in two proposals. He wants to choose the one that has higher average net present value and lower standard deviations. The relevant data are given below. Can you help him choosing the proposal?

Proposal A

Net present value (NPV) Chance of the possible outcome of NPV

1559 0.30

5662 0.40

9175 0.30

Proposal B

Net present value (NPV) Chance of the possible outcome of NPV

-10050 0.30

5812 0.40

20584 0.30

Explain, without using a truth table, why (p ∨ q ∨ r) ∧

(¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r

is true and at least one is false, but is false when all three

variables have the same truth value.

One of the city's professional football teams (team 1) plays at home and another (team 2) plays away on the same night. A professional football a 0.641 probability of winning a home game and a 0.462 probability of winning an away game. Historically, when both teams play on the same night, the chance that the next morning's leading sports story will be about the team 1 is 60 percent and the chance that it will be about the team 2 game is 40 percent. Suppose that on the morning after these games the newspaper's leading sports story begins with the headline ' we win!! ' what is the probability that the story is about team 1?

Determine whether each of these compound propositions

is satisfiable.

a) (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q)

b) (p → q) ∧ (p → ¬q) ∧ (¬p → q) ∧ (¬p → ¬q)

c) (p ↔ q) ∧ (¬p ↔ q)