Elementary Linear Algebra - Text Only (Looseleaf)
8th Edition
ISBN: 9781305953208
Author: Larson
Publisher: Cengage Learning
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Question
Chapter A, Problem 25E
To determine
To prove:
The set
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Describe the pictured subset of R? in two different ways, first using set-builder notation and
then as a Cartesian product of two sets.
(2,5)
(7,5)
(2,1)
(7,1)
Linear algebra
Relations
Given the sets over the domain {a, b, c, d}:
A = { (a, a), (b, b), (d, d), (b, a) }
B = { (a, c), (c, c), (c, a), (d, a) }
C = { (a, c), (a, d), (c, b), (c, c) }
For A, B, and C: find the following closures.
1. Reflexive
2. Symmetric
3. Transitive
Logic
1. Simplify the following using a truth table.
¬( (p ∧ q) -> (p ∨ q))
2. Show if the following is equivalent to ¬ p ∧ ¬ q . Use Boolean math. Label each law that you apply.
¬(p ∨ (¬ p ∧ q))
3. Simply the following using Boolean Math. Label each law that you apply.
b ∧ a ∨ e ∧ c ∧ ¬e ∨ ¬a ∧ b
4. Write out and prove (or disprove) the following statement.
“If I’m sleepy, then I’ll drink coffee or tea. I’ll never drink tea. Therefore, if I’m sleepy, then I'll drink coffee"
Convert the English sentence to an argument (short notation). Then, use a truth table to prove/disprove it:
Chapter A Solutions
Elementary Linear Algebra - Text Only (Looseleaf)
Ch. A - Using Mathematical Induction In Exercises 1-4, use...Ch. A - Prob. 2ECh. A - Prob. 3ECh. A - Prob. 4ECh. A - Prob. 5ECh. A - Prob. 6ECh. A - Prob. 7ECh. A - Prob. 8ECh. A - Prob. 9ECh. A - Prob. 10E
Ch. A - Prob. 11ECh. A - Prob. 12ECh. A - Prob. 13ECh. A - Prob. 14ECh. A - Using Proof by Contradiction In Exercises 1526,...Ch. A - Prob. 16ECh. A - Prob. 17ECh. A - Prob. 18ECh. A - Prob. 19ECh. A - Prob. 20ECh. A - Prob. 21ECh. A - Prob. 22ECh. A - Prob. 23ECh. A - Prob. 24ECh. A - Prob. 25ECh. A - Prob. 26ECh. A - Prob. 27ECh. A - Prob. 28ECh. A - Prob. 29ECh. A - Prob. 30ECh. A - Prob. 31ECh. A - Prob. 32ECh. A - Prob. 33E
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- Proof Prove that if S1 is a nonempty subset of the finite set S2, and S1 is linearly dependent, then so is S2.arrow_forwardDetermining Whether a Set Is a Basis In Exercises 35-38, determine whether the set {v1,v2}is a basis for R2.arrow_forwardProof Let S={u,v} be a linearly independent set. Prove that the set {u+v,uv} is linearly independent.arrow_forward
- Real Analysisarrow_forwardLet X be a superset of Y. Let Z be a set. Using an example of your own, state whether thefollowing are correct or incorrect. If incorrect, correct by changing the right hand side ofthe equation.(i) X n Y = Y(ii) X u Y = Ø (iii) X n Z ⊂ Y u Zarrow_forwardHow do you write the proof? Also how do you know what to write in the proof?arrow_forward
- PROOF EVALUATION (This type of exercise will appear occasionally): Each of the follow- ing is a proposed "proof" of a "theorem". However the "theorem" may not be a true statement, and even if it is, the "proof" may not really be a proof. You should read each "theorem" and "proof" carefully and decide and state whether or not the "theorem" is true. Then: • 18 If the "theorem" is false, find where the "proof" fails. (There has to be some error.) If the "theorem" is true, decide and state whether or not the "proof" is correct. If it is not correct, find where the "proof" fails.arrow_forwardMathmaticsarrow_forwardHow do you solve this question on discrete mathematics: 2. Let A be the set of all points in three-dimensional space. Define R as the relation that two points will have the same z coordinate. (a) Give three examples that are related to P(4,6,2). (b) Prove or disprove that R is reflexive. (c) Prove or disprove that R is symmetric. (d) Prove or disprove that R is transitive. (e) Is R an equivalence relation? (f) What space do all points related to each other (e.g. all points withz= 2) describe?arrow_forward
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