# 离散数学代写|discrect mathematics代写CMTH110代考

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This course covers the fundamentals of discrete mathematics with a focus on proof methods. Topics include: propositional and predicate logic, notation for modern algebra, naive set theory, relations, functions and proof techniques.

The number of elements of the set $X$, written $|X|$, is its cardinality.
From the definition, $|{1,2,3}|=3$. Also, $|\varnothing|=0$.
Definition 1.2 1. Given sets $X$ and $Y$, we say $X$ is a subset of $Y$ if every element of $X$ is also an element of $Y$. We write $X \subseteq Y$, and refer to the symbol $\subseteq$ as inclusion.

1. If there is an element of $Y$ that is not an element of $X$, we say that $X$ is a proper subset of $Y$, written $X \nsubseteq Y$. Note that if $X \varsubsetneqq Y$, then we also have that $X \subseteq Y$.
Example $1.2$ The set
$${1,3,4} \subseteq{1,2,3,4}$$
and it is also the case that
$${1,3,4} \varsubsetneqq{1,2,3,4}$$

These are two equations in two unknowns, and we solve them below
Subtracting (1.4) from (1.3), we obtain
$$V_{1}(H)-V_{1}(T)=\Delta_{0}\left(S_{1}(H)-S_{1}(T)\right)$$
so that
$$We may put sets together using several operations. Definition 1.6 Let X and Y be sets in a universe U. 1. The intersection of X and Y, written X \cap Y, is the set consisting of elements in both X and Y. We write$$
X \cap Y={u \in U: u \in X \text { and } u \in Y}
$$2. The union of X and Y, written X \cup Y, is the set consisting of elements in X or Y. We write$$
X \cup Y={u \in U: u \in X \text { or } u \in Y}
$$3. The difference of X with Y, written X \backslash Y, is the set consisting of elements in X but not in Y. We write$$
X \backslash Y={u \in U: u \in X \text { and } u \notin Y}
$$4. The complement of X, written X^{c}, is the set of elements of U not in X. We write$$
X^{c}={u \in U: u \notin X}


## CMTH110 COURSE NOTES ：

Definition 1.1. A statement (or proposition) is a sentence that is either true or false but not both.
Examples of statements:

• $p=$ “Alice is 12 years old”,
• $q=$ “Bob ate pizza for dinner yesterday”,
• $r=” 3 \cdot 3=9$ “,
• $s=” 2 \cdot 4=9 “$.
Examples of non-statements:
• “Is the sky blue?” (question)
• ” $x^{2}+y^{2}=13 “$ (for some values of $x$ and $y$ the proposition is true, whereas for others it is false).
Compound Statements
We can make new statement from old ones using operators like “not”, “and”, “or”, “if then”. There are a few simple examples:
• “Alice is not 12 years old” $(\sim p)$,
• “Bob ate pizza for dinner yesterday and $3 \cdot 3=9$ ” $(q \wedge r)$.
Compound statements (since they are (regular) statements as well) must have well-defined truth values – they must be either true or false. We can write down a table of truth values for compound statements based on the truth values of component statements.
Definition 1.2. The negation of a statement $p$ (denoted $\sim p$ ) is true if $p$ is false. If $p$ is true, then $\sim p$ is false.