Math 266 Final flashcards

X, or first variable.

Y, or second variable.

A proper subset of a set is a subset that is not equal to its containing set. Thus, A is a proper subset of B if A is in B and there is at least one element in B that is not in A.

f(A)

f^-1(A)

x is domain

y is co-domain

IX(x)=x for all x ∈ X

basically, whatever the input of the function is.

F is one-to-one (or injective) if, and only if any two distinct elements in X are sent to two distinct elements in Y.

A function f is onto if each element in Y has at least one corresponding x value

function that is both one to one and onto

If F is a one-to-one correspondence from a set X to a set Y , then there is a function from Y to X that “undoes” the action of F; that is, it sends each element of Y back to the element of X that it came from.

g circle f or g of f of x. g(f(x))

If f is a function from a set X to a set Y,and Ix is the identity function on X,and Iy is the identity function on Y,then f circle Ix = f. and Iy circle f = f

If F: X → Y and G: X → Y are functions, then F = G if, and only if, F(x) = G(x) for all x ∈ X.

composition of one to one functions is one to one

Composition of onto functions is onto.

Composition of bijections is a bijection

Inverse of a bijection is a bijection

p and q

p or q

A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables.

not p

Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables.

A contradication is a statement form that is always false regardless of the truth val- ues of the individual statements substituted for its statement variables

If p and q are statement variables, the conditional of q by p is “If p then q” or “p implies q” and is denoted p → q.It is false when p is true and q is false; otherwise it is true.

The contrapositive of p → q is ∼q → ∼p.

The converse of p → q is q → p

The inverse of p → q is ∼p → ∼q

p, if and only if, q. p ↔ q. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.

means “if r then s.”

means “if not r then not s.”

An argument is a sequence of statements

All statements in an argument and all statement forms in an argument form, except for the final one, are called premises (or assumptions or hypotheses)

To say that an argument form is valid means that no matter what particular state- ments are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true.

A row of the truth table in which all the premises are true is called a critical row. If there is a critical row in which the conclusion is false, then it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid. If the conclusion in every critical row is true, then the argument form is valid.

A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.

he domain of a predicate variable is the set of all values that may be substituted in place of the variable.

If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x.

The notation d|n is read “d divides n.” Symbolically, if n and d are integers and d ̸= 0:

Solution First find all the factors of 3,300. Then write them in ascending order:
3,300 = 100·33 = 4·25·3·11 = 2·2·5·5·3·11 = 2^3 ·3^1 ·5^2 ·11^1

Given any integer n and positive integer d, there exist unique integers q and r such
that

n=dq+r and 0≤r<d.

Any two consecutive integers have opposite parity. Even, odd