Set Builder Notation

If you recall, a set is a collection of objects which we write using brackets and name using a capital letter. Remember also that: means is an element of. means is NOT an element of. or { } means the empty set and contains no elements. Also a set is a subset of another if every element in one set is also an element in the other set. To write this, we use the notation

Lastly, remember that the intersection of two sets, or groups is the elements that both groups have in common. The word AND is used to mean an intersection. You can also use the symbol: Finally, the union of two sets, or groups is the combination of the elements in both groups. It doesn’t matter if they are in common or not. The word OR is used to mean a union. You can also use the symbol:

We can count the number of elements in a set. We use the notation n(A) to represent the number of elements in set A. For example: The set {1, 2, 3, 4, 5} has five elements in it, so we can write n(A) = 5. We read this as the number of elements in A is 5.

A finite set has a finite number of elements in it (a countable quantity). An infinite set has infinitely many elements in it (an uncountable quantity). For example: The set of all positive integers less than 100 is a finite set because we can count the exact number. The set of all integers is an infinite set because we can never count the exact number of integers total.

Set Builder Notation is a way that we can describe sets using symbols. It looks complicated but it is a way to make our work easier. means “such that” means “the set of” means “also” So, we get a statement that looks like: which we translate to: A is the set of all x such that x is an element of the integers, also x is between -2 and 4, including -2 and 4.

Translate the following statements: B is the set of all x such that x is an element of the real numbers, also x is between 0and 1, including 0 and 1. A is the set of all x such that x is an element of the integers, also x is greater than 0. G is the set of all x such that x is an element of the integers, also x is between 1and 2, including 1and 2. D is the set of all X such that x is an element of the Rational Numbers, also x is less than -1, including -1.

State if the following sets are finite or infinite: B is an infinite set since it includes all real numbers between 0 and 1. A is an infinite set since it includes all integers greater than 0. G is a finite set since there are only two numbers that work. They are 1 and 2.

Given that: write down: a) the meaning of the set in words. b)the elements that are found in set A. c) n(A). Answer: a) A is the set of all x such that x is an integer, also is between 3 and 10 including 10. b) A = {4, 5, 6, 7, 8, 9, 10} c) n(A) = 7 Given that: write down: a) the meaning of the set in words. b)the elements that are found in set B. c) n(B). Answer: a) B is the set of all x such that x is an integer, also x is between -1 and 4. b) B = {0, 1, 2, 3} c) n(B) = 4

Given that: write down: a) the meaning of the set in words. b)the elements that are found in set A. c) n(A). Answer: a) A is the set of all x such that x is an rational number, also is between 1 and 3 including 3. b) Impossible to list since this is an infinite set. c) n(A) = infinite Given that: write down: a) the meaning of the set in words. b)the elements that are found in set B. c) n(B). Answer: a) B is the set of all x such that x is an natural number, also x is between -3 and 3. b) B = {0, 1, 2} c) n(B) = 3

Given that: and is Answer: Yes A is a subset of B since every element of A is also in B. Given that: and is Answer: Yes A is a subset of B since every element of A is also in B.

Universal Sets – The symbol U is used to represent a universal set. The main set for the given situation (subsets will be selected from this set). Complementary Sets – The complement of A, with symbol A’, is the set of all elements of U which are not in A. This is used to represent everything that is not in a particular set.

The relationships connecting A and A’ are:

Here is an example: Given that and, write down A’. Answer: A’ = {2, 4, 6} Given that and, write down A’. Answer: A’ = {all odd integers}

Given that and, write down A’. Answer: Given that and, write down A’. Answer: A’ = {all irrational numbers}

Given that and and list the elements of the following sets: a)A b)B c)A’ d)B’ e) f) g) h) Answer:

Given that and and list the elements of the following sets: a)A b)B c)A’ d)B’ e) f) g) h) Answer:

Given that and and find the following: a)A b)B c) d) e) Verify that Answer: