Second flshcards Flashcards
Writing Sets: Set-Builder Notation
The second way to show a set is called set-builder notation. It describes the properties an element must have to be included in the set, rather than listing all of the individual elements. It names a variable to represent potential elements of the set and defines the rule that determines whether it is an element. For example, the set S containing the factors of 10 would be written in set-builder notation as S={x|x is a factor of 10} . The vertical line after the first “x” is read as “such that,” so you would read this as, “the set of all real numbers x, such that x is a factor of 10.” The set E containing even numbers would be written as E={x|x is an even number} , which would be read as “the set of all real numbers x, such that x is a factor of 10.”
Subsets
A subset contains elements from a set. We use the notation T⊂S to indicate that set T is a subset of set S . For example, the set of natural numbers is a subset of the set of integers, so we could write N⊂Z . If a universal set has been established, then all other sets are considered to be its subsets. Equal sets are considered subsets of each other, and the empty set is considered a subset of all sets. When one set A contains all elements of another set B and other elements as well, we say that B is a proper subset of A .
Complement of a Set
The complement of a set is all elements that are part of the universal set but NOT part of that set. For example, if the universal set is the set of integers and set S is the set of negative integers, the complement of set S , written as Sc or S′ , would contain zero and all positive integers. The set of irrational numbers is usually written as Q′ because it is defined as all real numbers that are not rational.
Intersection of Two Sets
The intersection of two sets is the set of elements that the two sets have in common. For two sets S and T , the intersection is written as S∩T . For example, if S={x|x is a prime number} and T={x|x<12} , then S∩T={2,3,7,11} , which is the set of all prime numbers that are also less than 12.
Union of Two Sets
The union of two sets is the set of all elements in both sets combined. For two sets S and T , the union is written as S∪T . For example, let’s say that S is the set of the factors of 12: S={1,2,3,4,6,12} and T is the set of factors of 8: T={1,2,4,8} . There is no need to repeat elements belonging to both sets when writing their union, so we would write the union of these two sets simply as S∪T={1,2,3,4,6,8,12} .
Infinite Sets
Some sets have only a certain number of elements, while other sets are infinite. Almost all sets in the number system including real numbers, integers, rational numbers, natural numbers, and prime numbers are infinite sets. It may seem strange that those sets are all infinite, since some are clearly larger than others; for example, the integers set is clearly larger than the prime numbers set. Nonetheless, all of these sets are considered infinite.
Compound Statements
To build a compound statement, we need to introduce a second simple statement that can be combined with the first one. We typically use the letter q to represent a second simple statement in logic notation. For example, we could say “q: Jeremy has blue eyes.” This is also a simple statement because it is either true or false.
Conditional Statements
Conditional statements use the words “if” and “then” to explain how the truth value of one simple statement affects the truth value of another. For example, consider the following two simple statements:
pq:: We’re having a party. We will buy food.
Using the two simple statements, you can form the conditional statement “IF we’re having a party, THEN we will buy food.” In logic notation, this is written as p→q and read as “if p, then q.” We call p the antecedent of the statement and q the consequent. A conditional statement is considered false only when the antecedent is true and the consequent is false. All other combinations of truth values make the conditional true.
Logical Arguments
Logical arguments consist of premises and conclusions. For example, take the argument “All chairs are furniture, and an armchair is a chair. Therefore, an armchair is furniture.” The statements “All chairs are furniture” and “An armchair is a chair” are the premises, and “An armchair is furniture” is the conclusion. Assuming the premises are true, the argument is valid if the conclusion follows logically from the premises.
Ray
A line with one endpoint, so that it goes on forever in only one direction is called a ray:
line with a circle on one end and an arrow on the other end
Line Segment
A line with two endpoints is called a line segment:
Parallel Lines
Two lines that will never intersect or cross no matter how long you extend them for are called parallel lines:
two lines that do not intersect with arrows on either end
Right Angle
A 90-degree angle is also called a right angle, and lines that intersect at right angles are called perpendicular lines. Right angles are often indicated with a little square as shown here:
two lines intersecting with a square in the middle showing a right angle
Angles and Their Properties
When two lines, rays, or segments cross each other, they are called intersecting lines, rays, or segments. When lines intersect, they form angles. The following image shows two intersecting lines with curves marking 4 angles that they form with each other:
two lines intersecting to form angles
Angles are typically measured in degrees. A full circle is 360 degrees (360°), so the degree measure of an angle is the portion of a circle it covers times 360°.
For example, an angle that forms a straight line (half a circle) is 180° because 180 is half of 360, and an angle that forms an L-shape (quarter circle) is 90° because 90 is a quarter of 360. If two angles have the same measure, we say they are congruent.
Parallel Lines Cut by a Transversal—Properties
A lot of interesting relationships between angles emerge when we cut across two parallel lines with a third line called a transversal as shown here:
two parallel lines with one line intersecting both of them and the angles
The first thing we might notice here is that, based on what we already know, the angle pairs A and B, A and D, B and C, D and C, and then E and F, E and H, F and G, and G and H are all supplementary, meaning their measures add up to 180°.
Because the two lines are parallel, the transversal must form the same angles with both of them. This means that alternate exterior angles are congruent. In this diagram, angles A and G and angles B and H are both pairs of alternate exterior angles.
For the same reason, alternate interior angles are also congruent. In this diagram, angles D and F and angles C and E are both pairs of alternate interior angles in this diagram.
Angle pairs like A and C, D and B, E and G, and F and H are called vertical angles and are also congruent.
Taken together, this also means that same-side interior angles are supplementary. Angles E and D and angles F and C are both pairs of same-side interior angles.
