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National Board: Mathematics EA > Context for Mathematics: Statements > Flashcards

Context for Mathematics: Statements Flashcards

1
Q

A statement, proposition, or assumption that precedes a conclusion, in an argument.

A

Premise

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2
Q

A conclusion made from two more more premises.

A

Argument

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3
Q

The following is an example of…
“If it is hot, then I will go swimming.
It is hot.
Therefore, I will go swimming.”

A

Premise and Argument
(Statements and Conclusions)

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4
Q

Any sentence or expression that has a truth value – that is may be considered true or false.

A

Statement

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5
Q

Type of statement that cannot be broken down into smaller statements. Also known as a proposition.

A

Simple Statement

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6
Q

The following are examples of a…
“It rains tomorrow.
The streets will be flooded.
Traffic will be slow.”

A

Simple Statement

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7
Q

A statement that contains one or more other statements, combined or modified in some way.

A

Compound Statement

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8
Q

The following is an example of a…
“If it rains tomorrow, then the streets will be flooded and traffic will be slow.”

A

Compound Statement

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9
Q

Common operations used to combine or modify simple statements.
Hint: there are 4.

A

Conjunction
Disjunction
Negation
Implication

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10
Q

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.

A

Conjunction

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11
Q

The following is an example of a…
“Roses are red and violets are blue.”

A

Conjunction

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12
Q

Represented by the operator ∧ or the word “and”

A ∧ B

A

Conjunction

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13
Q

An operation that combines two or more statements that is true if and only if at least one of the components is true.

A

Disjunction

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14
Q

The following is an example of a…
“I’ll clean the house today, or you’ll clean it tomorrow.”

A

Disjunction

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15
Q

Represented by the operator ∨ or the word “or”

A ∨ B

A

Disjunction

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16
Q

Operation that includes the possibility of both propositions, meaning and and or both make the statement true.

A

Disjunction

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17
Q

Compound statement of the form
“if P, then Q.
P implies Q.
P only if Q.
P → Q”
Also called implication.

A

Conditional Statement
(Implication)

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18
Q

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.”

A

Conditional Statement (Implication)

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19
Q

Combination of two conditional statements, where either both are true or both are false.

A

Biconditional statement

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20
Q

A compound statement in the form
“if P, then Q AND if Q, then P”
P if and only if Q
P ↔ Q”

A

Biconditional Statement

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21
Q

The following is an example of a…
“A number is even if and only if it is divisible by 2.”

A

Biconditional Statement

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22
Q

A logical construct used in a compound statement to give information on the number of subjects for which a statement is true.

A

Quantifier

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23
Q

Specifies that the statement is true for at least one subject. Usually introduced as “there exists.”
Representing by the symbol, ∃.

A

Existential Quantifier

24
Q

The following is an example of…
“There exists a real number that is equal to its own square.

A

Existential Quantifier

25
Q

Specifies that a statement is true for every subject. Usually said as “all” or “for all”. Represented by the symbol, ∀.

A

Universal Quantifier

26
Q

The following is an example of…
“All real numbers are less than their own squares plus one.”

A

Universal Quantifier

27
Q

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.”

A

Exclamation Mark after Existential Quantifier
∃!

28
Q

Shows the truth value of one or more compound statements for each possible combination of truth values of the propositions within it.

A

Truth Tables

29
Q

The truth table contains one column for each __________ & _________ _________.

A

Proposition
Compound Statements

30
Q

The truth table contains one row for each __________ of _________ _________.

A
31
Q

In a truth table, the expression that represents the number of possible combinations of N propositions.

A

2^n

32
Q

The Rule of ____________ states that for every case where (P → Q) ∧ P is true, Q will also be true.

A

Rule of Detachment

33
Q

The _________ Rule states that for every case where (P → Q) ∧ (Q → R), P → R will also be true.

A

The Chain Rule

34
Q

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.”

A

Converse

35
Q

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.”

A

Inverse

36
Q

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.”

A

Contrapositive

37
Q

Refers to the inversion of the truth value of a statement.
P and ¬P

A

Negation

38
Q

A set of useful relations connecting negation, conjunction, and disjunction.

A

De Morgan’s Laws

39
Q

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)

A

De Morgan’s Laws

40
Q

The four types of reasoning.

A

Inductive
Deductive
Formal
Informal

41
Q

Method used to make a conjecture based on patterns and observation. The conclusion may be true or false.

A

Inductive Reasoning

42
Q

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.”

A

Inductive Reasoning

43
Q

Method that proves a hypothesis or set of premises. The conclusion will be true, given the premises are true. Utilizes logic to determine conclusion.

A

Deductive Reasoning

44
Q

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”

A

Deductive Reasoning

45
Q

Method that involves justification using steps and processes to arrive at a conclusion, writing proofs and using logic.

A

Formal Reasoning

46
Q

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.”

A

Formal Reasoning

47
Q

Method that involves patterns and observations to make conjectures. Examples may show possible conclusions and may not reveal a certain conclusion.

A

Informal Reasoning

48
Q

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.

A

Informal Reasoning

49
Q

Serve to show the deductive or inductive process that relates the steps leading to a conclusion.

A

Proof

50
Q

The purpose of this proof is to show that the conclusion is true, given that the hypothesis is true.

A

Direct Proof

51
Q

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.”

A

Direct Proof

52
Q

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.

A

Indirect Proof
(Proofs by Contradiction)

53
Q

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. “

A

Indirect Proof
(Proof by Contradiction)

54
Q

The purpose of this proof is to show that the negation of Q will yield the negation of P.

A

Proof by Contraposition

55
Q

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.”

A

Proof by Contraposition

56
Q

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.

A

Proof by Contradiction.

National Board: Mathematics EA (3 decks)

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