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GROUP holds four properties simultaneously: i) Closure, ii) Associative, iii) Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers R are all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.

GROUP holds four properties simultaneously: i) Closure, ii) Associative, iii) Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers R are all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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GROUP holds four properties simultaneously: i) Closure, ii) Associative, Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers Rare all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.
Transcribed Image Text:GROUP holds four properties simultaneously: i) Closure, ii) Associative, Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers Rare all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.
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