This is a note collection for MAST10005 (Calculus 1) from the University of Melbourne.
1.1 Mathematical Statements
- Mathematical Statement: a sentence or expression that is unambiguously true of false.
- e.g. let p be the statement1+1=3, p is an valid statement. p is false.
- counter case: f(x) is continuous is not a M.S. without further information.
- Conjunction: true for p and q. (let p and q be M.S.) Denoted as p \wedge q
- Disjunction: true for p or q. (let p and q be M.S.) Denoted as p \vee q
- Negation: true for not p. (let p be M.S.) Denoted as ~ p
- Condition: let D be a set. Condition is a statement that is true or false depending on the choice of x \in D. Denoted with notation p(x).
- e.g. Consider x \in N. Let p(x) be the condition “x is even”
- Existential Quantifier: Let D be a set and p(x) be a condition over D. When at least onex \in D such that p(x) is true we say “there exists x \in D such that p(x)”. Denoted as \exists x \in D \; p(x)
- Universal Quantifier: Let D be a set and p(x) be a condition over D. When p(x) is true for every x \in D we say “for all x \in D p(x)”. Denoted as \forall x \in D \; p(x)
- Set of Numbers:
- \mathbb{N} set of natural numbers. {1, 2, 3, 4, …}
- \mathbb{Z} set of integers.
- \mathbb{Q} set of rational numbers
- \mathbb{R} set of real numbers
- Set: A collection of unique objects.
- e.g. {0, 0, 0, 1, 2, 7, 9} = {0, 1, 2, 7, 9}
- Let A and B be sets. A and B are equal when every element of A is an element of B and every element of B is an element of A. Denoted A = B
- Expressing Sets: Let D be set and let p(x) be a condition over D. The set of elements of D for which p(x) is true is denoted {x \in D | p(x)} (set-builder notation)
- Subset: Let A and B be sets. When every element of A is an element of B, A is a subset of B. Denoted A \sqsubseteq B
- Union: Let A and B be subsets of a set U. Union of A and B is the set of elements that are in at least one of A and B. Denoted A \cup B. That is: x \in A \cup B \; if \; and \; only \; if \; x \in A \; or \; x \in B
- Intersection: Let A and B be subsets of a set U. Intersection of A and B is the set of elements that are both in A and B. Denoted A \cap B. That is: A \cap B = \{x \in A \cup B | x \in A \; and \; x \in B\}
- Empty Set: \emptyset = \{\}
- Cartesian Product: Let A and B be sets. Cartesian product of A and B is the set of all poosible ordered pairs we can build using elements of A as the first element and elements of B as the second element. Denoted as A \times B = \{(a, b)|a \in A, b \in B\}
- e.g. \{0,1\} \times \{u,v,w\} = \{(0,u),(0,v),(0,w),(1,u),(1,v),(1,w)\}
- Function: A function f consists of:
- A nonempty set A called the domain of f;
- A nonempty set B called the codomain of f;
- A subset of AxB such that each element of A appears as the first element in exactly one ordered pair.
- f with domain A and codomain B is denoted as f:A \longrightarrow B
- The image of a under f is given by the notation f(a)
- Range: Let A and B be sets and let f:A \longrightarrow B. The range of f is the set of values the function takes. Formally:
- range(f) = \{b \in B \; | \; there \; exists \; a \in B \; such \; that \; f(a) = b\}
- range(f) = \{f(a) \; | \; a \in A\}
- Image: Let A and B be sets and let f:A \longrightarrow B. Let S be a subset of A. Image of S under f is the set
- f(S) = \{ b \in B \; | \; there \; exists \; s \in S \; such \; that \; f(s) = b \}
- f(S) = \{ f(s) \; | \; s \in S \}
- Image is generalisation of range.
- Function: Let A be a subset of R. Letf : A \longrightarrow \mathbb{R} and g : A \longrightarrow \mathbb{R}. Define following functions with codomain equal to R:
- (f+g)(x)=f(x)+g(x)
- (f-g)(x)=f(x)-g(x)
- (fg)(x)=f(x)*g(x)
- (f/g)(x)=f(x)/g(x)
- Domain for f+g, f-g and fg is set A. Domain for f/g is set \{ a \in A \; | \; g(a) \neq 0 \}