MATH 1332 | Units 1 and 2 Concepts

Remember: These flashcards alone will not teach you everything you need to know. You need to do the practice by applying these concepts. Answer with Term. I suggest using the Learn feature; start with multiple choice and T/F to get accustomed to the flashcards, then use the flashcards.

Statement

A sentence that is either true or false, but not both simultaneously.

Commands, questions, or opinions.

Statements are not…

Simple statement

A statement that conveys one idea and has no connectives linking it with another statement.

Compound statement

A statement that has two or more simple statements joined with two or more connectives.

Connectives

Words used to join simple statements. (Ex: and, or, if...then, and if and only if.)

Quantifiers

The words all, some, and no (or none).

Universal quantifiers

Words used in statements that either include or exclude every element of the universal set.

• All, each, every, no, none

Existential quantifiers

Words used in statements that claim the existence of something, but don't include the entire universal set.

• Some, there exists, at least one

Statement negation

Where a statement has the opposite meaning and truth value from the given statement.

• Represented by ~

• Not/no.

• A true statement becomes a false statement, and vice versa.

• The least dominant connective

The statements: Some A are not B

Or, Not all A are B.

negates…

The statements: All A are B

Or, There are no A that are not in B.

negates…

The statements: No/None of A are B

Or, All A are not B.

negates…

The statements: Some A are B

Or, There exists at least one A that is a B.

negates…

Conjunction

A compound statement that is true only when both simple statements are true.

• Represented as ∧

• And

• p and q

• A 3rd most dominant connective

Conjunctions in a truth table are…

Always false, except when T∧T and is true.

And in statements…

Refers to the conjunction (∧) of compound statements.

Disjunction

A compound statement that is false only when both component statements are false.

• Represented by ∨

• Or

• p or q or both

• A 3rd most dominant connective

Disjunctions in truth tables are…

Always true, except when F∨F and false.

Or in statements…

Refers to the disjunction (∨) of compound statements.

Conditional statements

A compound statement that is false only when the antecedent is true and the consequent is false.

• Represented by →

  • Is only true/good one way (→), and will be false the other way.

• If... then.

• "If p, then q" or “p implies q”

• -Antecedent → Consequent

• The 2nd most dominant connective

Conditionals in truth tables are…

Always true, except when T→F and is false.

Antecedent

The statement in a compound statement that is before the →.

Consequent

The statement in a compound statement that is after the →.

Biconditional statements

A compound statement that is true only when the component statements have the same truth value.

• Represented by ⟷

  • Is true/good both ways

• If and only if. (abbr. iff)

• "p if and only if q"

• The most dominant connective

Biconditionals in truth tables are…

Always false, except when T⟷T or F⟷F and is true.

the statements in parentheses appear on the same side of the comma.

• Ex: If q and ~p, then ~r

  • (q ∧~p) → ~r

In English simple statements,

Dominance of Connectives

If a symbolic statement does not have parentheses, statements before and after the most dominant connective should be grouped.

• Biconditional>Conditional>Conjunction and Disconjuntion> Negation

Tautology

A compound statement that is always true.

Equivalent compound statements

Two or more compound statements that are made up of the same simple statements and have the same corresponding truth values for all true-false combinations.

then its equivalent statement must be true.

If a compound statement is true,

then its equivalent statement must be false.

If a compound statement is false,

The two parts of an argument

• Premsies/assumptions

• Conclusion

Premisies/Assumptions

The given statements in an argument.

Conclusion

The result determined by the truth of the premises within an argument.

Valid argument/Tautology

The conclusion is true whenever the premises are assumed to be true.

Invalid argument/Fallacy

An argument that is not valid, or has at least one false in its truth table.

[(premise1)∧(premise2)∧...∧(premise n)] → conclusion, where n is the number of premises.

The symbolic form of an argument and its conclusion

Truth table basic structure

p	q	Symbolic Statement
T	T
T	F
F	T
F	F

Set

A collection of objects whose contents can be clearly determined.

Elements/members

The objects in a set.

Word description

• A method for representing sets

• Can designate or name a set

• Ex: Set W is the set of days of the week.

Roster method

• A method for representing sets

• Lists the members of a set

• Ex: W= {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Set-builder notation

• A method for representing sets

• Explains the properties that its members must satisfy

• Ex: W = {x | x is a day of the week}

Empty/null set

• The set that contains no elements.

• Is a subset of every set.

  • Is ⊆ B, for any set B.

  • Is ⊂ B, for any set B other than the empty set.

{ } or Ø

Notation used to indicate that the set contains no elements.

∈

• Notation used to indicate that an object is an element of a set.

• It replaces the words, "is an element of."

∉

• Notation used to indicate that an object is not an element of a set.

• It replaces the words, "is not an element of."

The set of natural numbers

N={1, 2, 3, 4, 5, ...}

Ellipsis

Three dots after an element indicate an endless list.

≤ and < 

"Less than or equal to" and "Less than"

≥ and >

"Greater than or equal to" and "Greater than"

Cardinal number

• The number of set A is the number of distinct elements in set A.

• Represented by n(A), read "n of A."

• Is not changed by repeating elements

Equivalent set

• Set A and Set B contain the same number of elements.

• n(A) = n(B).

• If a set is equal, it is automatically equivalent.

then A is equivalent to B: n(A) = n(B).

If set A and set B can be placed in a one-to-one correspondence,

then A is not equivalent to B: n(A) ≠ n(B).

If set A and set B cannot be placed in a one-to-one correspondence,

Finite set

• A set whose cardinality is 0 [n(A) = 0, A is the empty set] or a natural number.

• Has a beginning and an end.

Infinite set

• A set whose cardinality is not 0 or a natural number.

• Has a beginning but no end.

Equality of sets

• Set A and Set B contain exactly the same elements, regardless of the order or repetition of elements.

• Expressed as A = B.

• If a set is equal, it is automatically equivalent.

Subset of a set

• If every element in set A is also an element in set B.

• A ⊆ B.

• Every set is a subset of itself.

Not a subset of a set

• If there is at least one element of set A that is not an element of set B.

• A ⊄ B.

Proper subset

• If set A is a subset of set B and sets A and B are not equal (A≠B).

• Expressed as A ⊂ B.

Number of subsets

• The number of distinct subsets of a set

• n elements, 2ⁿ.

• As we increase the number of elements in the set by one, the number of subsets doubles.

Number of proper subsets

The number of distinct proper subsets of a set with n elements is 2ⁿ - 1.

Universal set

• A general set that contains all elements under discussion.

• U

• Represented by a rectangle

• Subsets within the universal set are depicted by circles, or sometimes other shapes.

John Venn

• 1843-1923

• Created Venn Diagrams to show the visual relationship among sets.

Disjoint sets

Two sets that have no elements in common.

Proper subsets

All elements of set A are elements of set B.

Equal sets

If A = B, then A ⊆ B and B ⊆ A.

Sets with some common elements

If set A and set B have at least one element in common, then the circles representing the sets must overlap.

Complement of a set

• The set of all elements in the universal set that are not in A. Symbolized by A'.

• A ' = {x | x ∈U  and x∉A}.

• If you add it to its set, you get the universal set.

Intersection of sets

• The set of elements common to both set A and set B.

• And

• A∩B =  {x | x∈A  and x∉B}.

• A∩B

Union of sets

• The set of elements that are members of set A or of set B or of both sets.

• A∪B.

• A∪B =  {x | x∈A  or x∈B}.

• Or

A

A ∪ Ø = ?

Ø

A ∩ Ø = ?

always begin by performing any operations inside the paretheses.

When performing set operations with parentheses,

its best to start with the innermost part and move outward.

When filling a Venn diagram,

Or in sets…

Refers to the union of sets.

And in sets…

Refers to the intersection of sets.

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

The cardinal number of the union of two finite sets

p ∧ q

p and q

p ∧ q

p but q

p ∧ q

p yet q

p ∧ q

p nevertheless q

p ∨ q

p or q

p → q

If p then q

q ← p

q if p

p → q

p is sufficient for q

p → q

q is necessary for p

p → q

p only if q

p → q

Only if p, q.

~p ← ~q

This is not p if it is not q

q ← ~p

q if not p

p ← ~q

p is necessary for not q

p → ~q

p is sufficient for not q

~q → p

no q only if p

p ⟷ q

p if and only if q

p ⟷ q

q if and only if p

p ⟷ q

If p then q, and if q then p

p ⟷ q

p is neccessary and sufficient for q