Context for Mathematics: Statements Flashcards
A statement, proposition, or assumption that precedes a conclusion, in an argument.
Premise
A conclusion made from two more more premises.
Argument
The following is an example of…
“If it is hot, then I will go swimming.
It is hot.
Therefore, I will go swimming.”
Premise and Argument
(Statements and Conclusions)
Any sentence or expression that has a truth value – that is may be considered true or false.
Statement
Type of statement that cannot be broken down into smaller statements. Also known as a proposition.
Simple Statement
The following are examples of a…
“It rains tomorrow.
The streets will be flooded.
Traffic will be slow.”
Simple Statement
A statement that contains one or more other statements, combined or modified in some way.
Compound Statement
The following is an example of a…
“If it rains tomorrow, then the streets will be flooded and traffic will be slow.”
Compound Statement
Common operations used to combine or modify simple statements.
Hint: there are 4.
Conjunction
Disjunction
Negation
Implication
An operation that combines two or more statements into a compound statement that is true if and only if all of the components are true.
Conjunction
The following is an example of a…
“Roses are red and violets are blue.”
Conjunction
Represented by the operator ∧ or the word “and”
A ∧ B
Conjunction
An operation that combines two or more statements that is true if and only if at least one of the components is true.
Disjunction
The following is an example of a…
“I’ll clean the house today, or you’ll clean it tomorrow.”
Disjunction
Represented by the operator ∨ or the word “or”
A ∨ B
Disjunction
Operation that includes the possibility of both propositions, meaning and and or both make the statement true.
Disjunction
Compound statement of the form
“if P, then Q.
P implies Q.
P only if Q.
P → Q”
Also called implication.
Conditional Statement
(Implication)
The following are examples of a…
“If a snake has rattles, then it’s venomous.
That a snake has rattles implies that it’s venomous.
A snake rattles only if it’s venomous.”
Conditional Statement (Implication)
Combination of two conditional statements, where either both are true or both are false.
Biconditional statement
A compound statement in the form
“if P, then Q AND if Q, then P”
P if and only if Q
P ↔ Q”
Biconditional Statement
The following is an example of a…
“A number is even if and only if it is divisible by 2.”
Biconditional Statement
A logical construct used in a compound statement to give information on the number of subjects for which a statement is true.
Quantifier
Specifies that the statement is true for at least one subject. Usually introduced as “there exists.”
Representing by the symbol, ∃.
Existential Quantifier
The following is an example of…
“There exists a real number that is equal to its own square.
Existential Quantifier
Specifies that a statement is true for every subject. Usually said as “all” or “for all”. Represented by the symbol, ∀.
Universal Quantifier
The following is an example of…
“All real numbers are less than their own squares plus one.”
Universal Quantifier
Specifies when there exists exactly one subject that meets a given criterion.
The following is an example: “there exists a unique real number that has an absolute value of zero.”
Exclamation Mark after Existential Quantifier
∃!
Shows the truth value of one or more compound statements for each possible combination of truth values of the propositions within it.
Truth Tables
The truth table contains one column for each __________ & _________ _________.
Proposition
Compound Statements
The truth table contains one row for each __________ of _________ _________.
In a truth table, the expression that represents the number of possible combinations of N propositions.
2^n
The Rule of ____________ states that for every case where (P → Q) ∧ P is true, Q will also be true.
Rule of Detachment
The _________ Rule states that for every case where (P → Q) ∧ (Q → R), P → R will also be true.
The Chain Rule
A conditional statement with the premise and conclusion interchanged.
For example, “P → Q and Q → P.
If there is a key in the lock, then someone is home.
If someone is home, then there is a key in the lock.”
Converse
A conditional statement with the premise and conclusion both negated.
For example, “P → Q and ¬P → ¬Q.
If there is a key in the lock, then someone is home.
If there is not a key in the lock, then no one is home.”
Inverse
A conditional statement with the premise and conclusion both negated and interchanged.
For example, “P → Q and ¬Q → ¬P.
If there is a key in the lock, then someone is home.
If no one is home, then there is not a key in the lock.”
Contrapositive
Refers to the inversion of the truth value of a statement.
P and ¬P
Negation
A set of useful relations connecting negation, conjunction, and disjunction.
De Morgan’s Laws
The negation of a conjunction is the disjunction of the negations.
The negation of a disjunction is the conjunction of the negations.
¬(P∧Q)↔(¬P)∨(¬Q) and ¬(P∨Q)↔(¬P)∧(¬Q)
De Morgan’s Laws
The four types of reasoning.
Inductive
Deductive
Formal
Informal
Method used to make a conjecture based on patterns and observation. The conclusion may be true or false.
Inductive Reasoning
The following is an example of _________ reasoning.
“A cube has…
A square pyramid has…
A triangular prism has….
Thus, the sum of the numbers of faces and vertices, minus the number of edges, will always equal 2.”
Inductive Reasoning
Method that proves a hypothesis or set of premises. The conclusion will be true, given the premises are true. Utilizes logic to determine conclusion.
Deductive Reasoning
The following is an example of _________ reasoning.
“If a ding is a dong, then a ping is a pong.
If a ping is a pong, then a ring is a ting.
A ding is a dong.
Therefore, a ring is a ting.
P → Q
Q → R
P ∴ R”
Deductive Reasoning
Method that involves justification using steps and processes to arrive at a conclusion, writing proofs and using logic.
Formal Reasoning
The following is an example of _________ reasoning.
“If a quadrilateral has four congruent sides, it is a rhombus.
If a shape is a rhombus, then the diagonals are perpendicular.
A quadrilateral has four congruent sides.
Therefore, the diagonals are perpendicular.”
Formal Reasoning
Method that involves patterns and observations to make conjectures. Examples may show possible conclusions and may not reveal a certain conclusion.
Informal Reasoning
The following is an example of _________ reasoning.
“The sum of 1 and 1/2 is 1 1/2.
The sum of 1, 1/2, and 1/4 is 1 3/4.
The sum of 1, 1/2, 1/4, and 1/8 is 1 7/8.
Thus, it appears that as the sequence approaches infinity, the sum of the sequence approaches 2.
Informal Reasoning
Serve to show the deductive or inductive process that relates the steps leading to a conclusion.
Proof
The purpose of this proof is to show that the conclusion is true, given that the hypothesis is true.
Direct Proof
The following is an example of a _________ proof.
“Prove - if m divides a and m divides b, then m divides a + b.
Proof:
- Assume m divides and and m divides b.
- Thus, a equals the product of m and some integer factor, p, by the definition of division, and b equals the product of m and some integer factor, q, by the definition of division. According to substitution, a + b may be rewritten as (m x p) + (m x q). Factoring out the m gives m(p + q). Since m divides p + q, and p + q is and integer, according to the closure property, we have shown that m divides a + b, by the definition of division.”
Direct Proof
The purpose of this proof is to show that a hypothesis is false, given the negation of the conclusion, indicating that the conclusion must be true.
Indirect Proof
(Proofs by Contradiction)
The following is an example of a _________ proof.
“Prove - If 3x + 7 is odd, then x is even.
Proof:
- Assume 3x is odd and x is odd.
- According to the definition of odd, x = 2a + 1, where is a is an element of the integers.
- Thus, by substitution, 3x + 7 = 3(2a + 1) + 7, which simplifies as 6a + 10, which may be rewritten as 2(3a + 5). Any even integer may be written as the product of 2 and some integer, k. Thus, we have shown the hypothesis to be false meaning the conditional statement must be true. “
Indirect Proof
(Proof by Contradiction)
The purpose of this proof is to show that the negation of Q will yield the negation of P.
Proof by Contraposition
The following is an example of proof by ___________.
“Prove - If 5x + 7 is even, then x is odd.
Proof:
- Assume that if x is even, then 5x + 7 is odd.
- Assume x is even.
- Thus by the definition of an even integer, x = 2a. By substitution, 5x + 7 may be rewritten as the product of 2 and some factor, k. Thus, 5x + 7 is odd, by definition of an odd integer. So, when 5x + 7 is even, x is odd.”
Proof by Contraposition
The purpose of this proof is to show the negation of q will result in a false hypothesis, indicating that the conclusion of the statement, as written, must be true.
Proof by Contradiction.
