Practice Algebra as Structure in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The perspective that algebra is the systematic study of abstract mathematical structures and the operations defined on them.

Beyond numbers: what happens when ANY set has operations with certain properties?

Showing a random 20 of 50 problems.

Example 1

easy
Is matrix multiplication commutative in general?

Example 2

medium
Does the set {0,1,2,3}\{0,1,2,3\} form a group under addition modulo 4? Verify the four group axioms.

Example 3

easy
Is the set of integers closed under addition? (Is a+ba+b always an integer?)

Example 4

medium
Is the set {0,1,2}\{0,1,2\} under addition modulo 3 closed? Check 2+2โ€Šmodโ€Š32+2 \bmod 3.

Example 5

challenge
Define aโˆ—b=a+bโˆ’aba*b = a + b - ab on Rโˆ–{1}\mathbb{R} \setminus \{1\}. Show it forms a group; find the identity and the inverse of aa.

Example 6

challenge
Show that if aโˆ—b=a+b+aba*b=a+b+ab on {x:xโ‰ โˆ’1}\{x: x\ne-1\}, the element 00 is the identity and find the inverse of aa.

Example 7

hard
Is the set of 2ร—22 \times 2 invertible matrices over R\mathbb{R} a group under multiplication? Name it.

Example 8

medium
Why is 00 not invertible under multiplication in any field?

Example 9

medium
In a group with operation โˆ—*, simplify (aโˆ—b)โˆ’1(a*b)^{-1}.

Example 10

easy
Name the additive identity in Z\mathbb{Z}.

Example 11

easy
What is the identity element for addition? For multiplication?

Example 12

easy
What is the additive inverse of 7 in the integers?

Example 13

hard
Does matrix multiplication have the same structural properties as number multiplication? Check commutativity.

Example 14

easy
Are the integers closed under division? (Is a/ba/b always an integer for integers a,bโ‰ 0a,b\ne0?)

Example 15

medium
Does the set {1,2,3,4}\{1,2,3,4\} form a group under multiplication modulo 5?

Example 16

medium
In Z5\mathbb{Z}_5, find the multiplicative inverse of 22.

Example 17

medium
Does subtraction have an identity element?

Example 18

hard
In Z6\mathbb{Z}_6, identify all elements that are not units (have no multiplicative inverse).

Example 19

easy
What is the multiplicative inverse of 55 in Q\mathbb{Q}?

Example 20

easy
Which axiom does 5+3=3+55 + 3 = 3 + 5 illustrate?
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