Guided Proof Prove that A is idempotent if and only if A T is idempotent. Getting Started: The phrase “if and only if” means that you have to prove two statements: 1. If A is idempotent, then A T is idempotent. 2. If A T is idempotent, then A is idempotent. (i) Begin your proof of the first statement by assuming that A is idempotent. (ii) This means that A 2 = A . (iii) Use the properties of the transpose to show that A T is idempotent. (iv) Begin your proof of the second statement by assuming that A T is idempotent.
Guided Proof Prove that A is idempotent if and only if A T is idempotent. Getting Started: The phrase “if and only if” means that you have to prove two statements: 1. If A is idempotent, then A T is idempotent. 2. If A T is idempotent, then A is idempotent. (i) Begin your proof of the first statement by assuming that A is idempotent. (ii) This means that A 2 = A . (iii) Use the properties of the transpose to show that A T is idempotent. (iv) Begin your proof of the second statement by assuming that A T is idempotent.
Solution Summary: The author explains that the matrix A is idempotent if and only when AT
Exercise 16: Write up three proofs for the above six theorems. (You should be able to do all of them, but write up three of them.)
Theorem 26: Suppose a, b ∈ Z. Then a ≤ b if and only if b − a ∈ N.
Theorem 27: Suppose a, b, c ∈ Z. If a < b then a + c < b + c. If a ≤ b then a + c ≤ b + c.
Theorem 28: Let a ∈ Z. Then a ∈ N if and only if a ≥ 0. Also, a is positive if and only if a > 0, and a is negative if and only if a 76 Chapter 4. The Integers Z
Theorem 29: Transitivity for < holds.
Theorem 30: Transitivity for ≤ hold. Mixed transitivity for < and ≤ hold. Theorem 31: Trichotomy for < holds.
For all sets A and B,
(A ∪ Bc) − B = (A − B) ∪ Bc.
An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.)
Let any sets A and B be given. Then
(A ∪ Bc) − B
=
(A ∪ Bc) ∩ Bc
by the set difference law
(1)
=
(Bc ∩ A) ∪ (Bc ∩ Bc)
by the distributive law
(2)
=
(Bc ∩ A) ∪ Bc
by the idempotent law for ∪
(3)
=
(A − B) ∪ Bc
by the set difference law
(4)
The commutative law is needed between between steps (1) and (2).
The complement law is needed between steps (2) and (3).
The commutative law is needed between between steps (3) and (4).
The absorption law is needed between steps (2) and (3).
The double complement law is needed between steps (3) and (4).
The rules of mathematical logic specify methods of reasoning mathematical statements. These rules are described either through predicate logic or propositional logic.
1- Define primes and describe why Goldbach’s Conjecture is classified as an open problem.
2- How can we describe gcd as a linear combination in terms of BE´ ZOUT’S THEOREM.
Chapter 2 Solutions
Bundle: Elementary Linear Algebra, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term
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