CSET SUBTEST 1 EXAM PREVIEW WITH
ANSWERS.
Group - Answers-A set of elements that is
1. Closed (combining any two elements produces a third element still in that set) under
one operation
2. Identity Element
3. Inverse Element
4. Associativity
Abelian Group - Answers-A group that includes the commutative property
Ring - Answers-A set of elements that is a group and is closed under 2 operations.
1. Closed under addition and multiplication
2. Associativity of addition and multiplication
3. Commutativity of addition
4. Identity Elements 0 and 1
5. Inverse Element for addition
6. Distributivity of addition and multiplication
Key Difference: No Multiplicative Inverse, No Commutative Multiplication
Examples: integers, square matrices, polynomials with coefficients in a ring
Field - Answers-A set of elements that is closed under addition and multiplication
1. Associativity of addition and multiplication
2. Commutativity of addition and multiplication
3. Identity Elements 0 and 1
4. Inverse Elements -a and a^-1
5. Distributivity
Examples: rational, real, and complex numbers
Modular Addition/Multiplication - Answers-a + b(mod n) = r
1. n is the number of elements in the set, including 0
2. r is the remainder when a + b or a*b is divided by n
3. Only applied if a + b or a*b results in a number outside the given set
4. If is NOT prime, its divisors will have no multiplicative inverse
Natural Numbers - Answers-No Decimals or Fractions, must be greater than 0
, 1, 2, 3, 4, ...
Whole Numbers - Answers-The natural numbers including 0
0, 1, 2, 3, 4, ...
Integers - Answers-Includes positive and negative natural numbers. Denoted by the
letter Z
-2, -1, 0, 1, 2, ...
Rational Numbers - Answers-Any number that can be represented as a fraction or
decimal. If a decimal, then it must either terminate or repeat. Denoted by the letter Q
4.32, 3/4, 1.3333....
Irrational Numbers - Answers-Numbers that cannot be expressed as fraction. A decimal
number that never ends and does not repeat. Denoted by the letter I
pi
Real Numbers - Answers-Set of all rational, irrational, integer, whole, and natural
numbers. Denoted by the letter R
Imaginary Numbers - Answers-Denoted by lower case i, where i is the square root of
negative 1
Complex Numbers - Answers-Numbers containing both a real and imaginary part.
Written in the form a+bi, where a is real and bi is imaginary
Prime Factorization - Answers-Any non-prime number can be written as a product of
prime numbers.
Complex Conjugate - Answers-A complex number with the sign of the imaginary portion
switched.
e.g. a+bi and a-bi are complex conjugates
Commutative Property - Answers-The property that says that two or more numbers can
be added or multiplied in any order without changing the result.
a+b=b+a and a*b=b*a
Associative Property - Answers-Changing the grouping of numbers will NOT change the
value.
ANSWERS.
Group - Answers-A set of elements that is
1. Closed (combining any two elements produces a third element still in that set) under
one operation
2. Identity Element
3. Inverse Element
4. Associativity
Abelian Group - Answers-A group that includes the commutative property
Ring - Answers-A set of elements that is a group and is closed under 2 operations.
1. Closed under addition and multiplication
2. Associativity of addition and multiplication
3. Commutativity of addition
4. Identity Elements 0 and 1
5. Inverse Element for addition
6. Distributivity of addition and multiplication
Key Difference: No Multiplicative Inverse, No Commutative Multiplication
Examples: integers, square matrices, polynomials with coefficients in a ring
Field - Answers-A set of elements that is closed under addition and multiplication
1. Associativity of addition and multiplication
2. Commutativity of addition and multiplication
3. Identity Elements 0 and 1
4. Inverse Elements -a and a^-1
5. Distributivity
Examples: rational, real, and complex numbers
Modular Addition/Multiplication - Answers-a + b(mod n) = r
1. n is the number of elements in the set, including 0
2. r is the remainder when a + b or a*b is divided by n
3. Only applied if a + b or a*b results in a number outside the given set
4. If is NOT prime, its divisors will have no multiplicative inverse
Natural Numbers - Answers-No Decimals or Fractions, must be greater than 0
, 1, 2, 3, 4, ...
Whole Numbers - Answers-The natural numbers including 0
0, 1, 2, 3, 4, ...
Integers - Answers-Includes positive and negative natural numbers. Denoted by the
letter Z
-2, -1, 0, 1, 2, ...
Rational Numbers - Answers-Any number that can be represented as a fraction or
decimal. If a decimal, then it must either terminate or repeat. Denoted by the letter Q
4.32, 3/4, 1.3333....
Irrational Numbers - Answers-Numbers that cannot be expressed as fraction. A decimal
number that never ends and does not repeat. Denoted by the letter I
pi
Real Numbers - Answers-Set of all rational, irrational, integer, whole, and natural
numbers. Denoted by the letter R
Imaginary Numbers - Answers-Denoted by lower case i, where i is the square root of
negative 1
Complex Numbers - Answers-Numbers containing both a real and imaginary part.
Written in the form a+bi, where a is real and bi is imaginary
Prime Factorization - Answers-Any non-prime number can be written as a product of
prime numbers.
Complex Conjugate - Answers-A complex number with the sign of the imaginary portion
switched.
e.g. a+bi and a-bi are complex conjugates
Commutative Property - Answers-The property that says that two or more numbers can
be added or multiplied in any order without changing the result.
a+b=b+a and a*b=b*a
Associative Property - Answers-Changing the grouping of numbers will NOT change the
value.
