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11: Relations
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/11%3A_Relations
4.E: Systems of Linear Equations (Exercises)
https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Arnold)/04%3A_Systems_of_Linear_Equations/4.0E%3A_4.E%3A_Systems_of_Linear_Equations_(Exercises)
3) $\begin{aligned} 2 x+y &=6 \\ y &=x+3 \end{aligned}$ In Exercises 7-18, solve each of the given systems by sketching the lines represented by each equation of the given system on graph paper, the...3) $\begin{aligned} 2 x+y &=6 \\ y &=x+3 \end{aligned}$ In Exercises 7-18, solve each of the given systems by sketching the lines represented by each equation of the given system on graph paper, then estimating the coordinates of the point of intersection to the nearest tenth. 12) $\begin{aligned} 4 x+3 y &=24 \\ y &=\dfrac{1}{7} x+5 \end{aligned}$ 20) $\begin{aligned} 3 x+6 y &=2 \\ y &=-2 y+2 \end{aligned}$ 28) $\begin{aligned} 9 x+4 y &=-3 \\ y &=-\dfrac{9}{4} x+6 \end{aligned}$
4.3: Section 3-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/04%3A_Direct_Proof/4.03%3A_Section_3-
The first line of the proof is the sentence “Suppose P.” The last line is the sentence “Therefore Q.” Between the first and last line we use logic, definitions and standard math facts to transform the...The first line of the proof is the sentence “Suppose P.” The last line is the sentence “Therefore Q.” Between the first and last line we use logic, definitions and standard math facts to transform the statement P to the statement Q. As our first example, let’s prove that if x is odd then x2 is also odd. (Granted, this is not a terribly impressive result, but we will move on to more significant things in due time.) The first step in the proof is to fill in the outline for direct proof.
5.4: Adding and Subtracting Polynomials
https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Arnold)/05%3A_Polynomial_Functions/5.04%3A_Adding_and_Subtracting_Polynomials
In this section we concentrate on adding and subtracting polynomial expressions, based on earlier work combining like terms in Ascending and Descending Powers.
12.0: Section 1-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/12%3A_Functions/12.00%3A_Section_1-
A function f from A to B (denoted as $f : A \rightarrow B)$ is a relation $f \subseteq A \times B$ from A to B, satisfying the property that for each $a \in A$ the relation f contains exactly on...A function f from A to B (denoted as $f : A \rightarrow B)$ is a relation $f \subseteq A \times B$ from A to B, satisfying the property that for each $a \in A$ the relation f contains exactly one ordered pair of form $(a, b)$ . The range of f is the set $\{f(a) : a \in A\} = \{b : (a, b) \in f\}$ . (Think of the range as the set of all possible “output values” for f . Think of the codomain as a sort of “target” for the outputs.)
11.4: Section 5-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/11%3A_Relations/11.04%3A_Section_5-
For example, it is probably obvious to you already that elements of $\mathbb{Z}_{n}$ obey the commutative laws $[a] + [b] = [b] + [a]$ and $[a] \cdot [b] = [b] \cdot [a]$ . To see why, suppose Al...For example, it is probably obvious to you already that elements of $\mathbb{Z}_{n}$ obey the commutative laws $[a] + [b] = [b] + [a]$ and $[a] \cdot [b] = [b] \cdot [a]$ . To see why, suppose Alice and Bob want to multiply the elements $[a],[b] \in \mathbb{Z}_{n}$ , and suppose $[a] = [a′]$ and $[b] = [b′]$ .
11.2: Section 3-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/11%3A_Relations/11.02%3A_Section_3-
Given any element $a \in A$ , the equivalence class containing a is the subset $\{x \in A : xRa\}$ of A consisting of all the elements of A that relate to a. Define a relation S on A by declaring \...Given any element $a \in A$ , the equivalence class containing a is the subset $\{x \in A : xRa\}$ of A consisting of all the elements of A that relate to a. Define a relation S on A by declaring $xSy$ if and only if for some $n \in \mathbb{N}$ there are elements $x_{1}, x_{2}, \cdots , x_{n} \in A$ satisfying $x R x_{1}, x_{1} R x_{2}, x_{2} R x_{3}, x_{3} R x_{4}, \cdots , x_{n-1} R x_{n}$ , and $x_{n} R y$ .
2.3: Biconditional Statements
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/02%3A_Logic/2.03%3A_Biconditional_Statements
The expression $P \Leftrightarrow Q$ is understood to have exactly the same meaning as $(P \Rightarrow Q) \wedge (Q \Rightarrow P)$ . According to the previous section, $Q \Rightarrow P$ is r...The expression $P \Leftrightarrow Q$ is understood to have exactly the same meaning as $(P \Rightarrow Q) \wedge (Q \Rightarrow P)$ . According to the previous section, $Q \Rightarrow P$ is read as "P if Q," and $P \Rightarrow Q$ can be read as "P only if Q."
2.0: Statement
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/02%3A_Logic/2.00%3A_Statement
In discussing a particular statement, such as "The function $f(x) = x^2$ is continuous," it is convenient to just refer to it as R to avoid having to write or say it many times. This is a sentence t...In discussing a particular statement, such as "The function $f(x) = x^2$ is continuous," it is convenient to just refer to it as R to avoid having to write or say it many times. This is a sentence that is true. (All multiples of 6 are even, so no matter which multiple of 6 the integer x happens to be, it is even.) Since the sentence P is definitely true, it is a statement.
6.2: Section 3-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/06%3A_Proof_by_Contradiction/6.02%3A_Section_3-
For example, in proving a conditional statement $P \Rightarrow Q$ , we might begin with direct proof and thus assume P to be true with the aim of ultimately showing Q is true. This proposition can be...For example, in proving a conditional statement $P \Rightarrow Q$ , we might begin with direct proof and thus assume P to be true with the aim of ultimately showing Q is true. This proposition can be reworded as follows: If r is a non-zero rational number, then r is a product of two irrational numbers. We know $\sqrt{2}$ is irrational, so to complete the proof we must show $\frac{r}{\sqrt{2}}$ is also irrational.
10.2: Section 3-
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/10%3A__Mathematical_Induction/10.02%3A_Section_3-
It is there- fore more “automatic” than the proof by contradiction that was introduced in Chapter 6. For the sake of contradiction, suppose not every $S_{n}$ is true. Let $k > 1$ be the smallest i...It is there- fore more “automatic” than the proof by contradiction that was introduced in Chapter 6. For the sake of contradiction, suppose not every $S_{n}$ is true. Let $k > 1$ be the smallest integer for which $S_{k}$ is false. Then $S_{k-1}$ is true and $S_{k}$ is false. This setup leads you to a point where $S_{k-1}$ is true and $S_{k}$ is false. We use proof by smallest counterexample. (We will number the steps to match the outline, but that is not usually done in practice.)

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