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11 GEN-MATH Q2 > [DEFINE] LOGIC: Terminologies > Flashcards

[DEFINE] LOGIC: Terminologies Flashcards

1
Q

‌Logic

A

a system of reasoning that allows inferences to be drawn from facts.

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2
Q

Proposition

A

a declarative sentence that is either true or false, but not both.

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3
Q

Conjunction

A

‘AND’ statement in mathematics. A conjunction of two statements is true only when both statements are true.

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4
Q

Disjunction

A

“OR” statement in mathematics. A disjunction is false if and only if both statements are false; otherwise it is true.

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5
Q

‌Negation

A

“NOT” statement in mathematics. It is a statement with the opposite truth value. If a statement is true, then its negation is false, and vice versa.

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6
Q

‌Conditional (or implication)

A

a logical compound statement in which p implies q. The statement is only false when p is true but q is false.

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7
Q

‌Biconditional

A

True when both have the same truth value.

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8
Q

‌Exclusive-or

A

The statement is only true if it has differnent truth values. P and q cannot be the same

EX: I am a dog or I am a cat

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9
Q

‌Truth Values of proposition

A

The truth value of a proposition is a determination of whether the proposition is true (T) or false (F). In classical logic, a proposition can only have one of these two truth values.

Proposition: “The sun rises in the east.”

This statement is true. Hence, its truth value is T (true).

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10
Q

Conditional Propositions

A

an “if-then” expression. It is false only when p is true and q is false, and is true in all other situations.

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11
Q

Converse of p-q

A

A conditional proposition in which the statement “p implies q” is “q implies p”. It is created by swapping the order of the hypothesis and conclusion.

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12
Q

Inverse of p-q

A

A conditional proposition in which the statement “p implies q” is “not p implies not q.” The order of the hypothesis and the conclusion remains the same, but they are both negated.

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13
Q

Contrapositive

A

A conditional proposition in which the statement “p implies q” is “not q implies not p.” It is formed by negating and swapping the hypothesis and the conclusion

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14
Q

Tautology

A

a proposition that is always true

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15
Q

Contradiction

A

a proposition that is always false

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16
Q

Contingency

A

It is neither a tautology nor a contradiction. In other words, it is a proposition that is neither true nor false.

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17
Q

‌Proof

A

a logical argument that shows a mathematical statement is true.

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18
Q

‌Premises

A

AKA the cause. a statement that is assumed to be true and is used as the basis for an argument.

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19
Q

Conclusion

A

AKA the effect. a statement that is reached by applying logical rules to a set of premises.

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20
Q

Number Theory

A

branch of pure mathematics mainly to study natural numbers and integers.

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21
Q

Counting

A

a subset of whole numbers that do not include zero

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22
Q

‌Whole

A

a subset of whole numbers that includes the number zero

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23
Q

‌Integers

A

a whole number (not a fractional number) that can be positive, negative, or zero.

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24
Q

Rational Numbers

A

are numbers that can be written as a fraction or a ratio.

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25
Q

‌Irrational Numbers

A

are numbers that can’t be written as a fraction or ratio.

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26
Q

Real Numbers

A

are numbers that include rational and irrational numbers.

27
Q

‌Imaginary numbers

A

it is the product of a real number and the imaginary unit i. When squared, imaginary numbers also give a negative result.

28
Q

‌Complex numbers

A

these are numbers that have two parts: a real part and an imaginary part. Imaginary numbers are denoted by i.

*** it comes in the form a + ib

29
Q

Odd numbers

A

these numbers cannot be equally divided into pairs. They always end with the last digits 1, 3, 5, 7, 9.

30
Q

Even Numbers

A

any number divisible by 2. They always end with the last digits 0, 2, 4, 6, 8.

31
Q

Prime numbers

A

a number that can only be divided by itself and 1 without remainders. It is not a product of two smaller natural numbers.

32
Q

Composite Numbers

A

numbers that have more than two factors. It can be formed by multiplying two smaller positive integers.

33
Q

Rule of Inference

A

a logical form or guide consisting of premises (or hypotheses) and draws a conclusion.

34
Q

Conjecture

A

a mathematical statement which appears to be true, but has not yet been rigorously proved.

35
Q

Axiom (or postulate)

A

a statement that is assumed to be true without the need for proof.

36
Q

‌Theorem

A

a statement that has been or can be proven to be true based on known facts and mathematical operations

37
Q

‌Lemma (or pre-theorem)

A

a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem

38
Q

‌Corollary (or post-theorem)

A

a result in which the proof relies heavily on a given theorem

39
Q

‌Proof

A

a logical argument that shows a mathematical statement is true.

40
Q

Divisibility

A

this means a number can be divided without leaving a remainder

41
Q

Greatest Common Divisor

A

is the greatest value that can divide two integers evenly.

42
Q

‌Least Common Multiple

A

the smallest positive multiple that two or more numbers share

43
Q

Relatively Prime Integers

A

Two integers that have no common factors other than 1. This means that no other integer could divide both numbers evenly. Two integers a,b are called relatively prime to each other if gcd(a,b)=1

44
Q

‌Direct Proof

A

A method that involves proving a statement by assuming the hypothesis is true and demonstrating that the conclusion logically follows.

45
Q

Proof of Contrapositive (or proof of transposition)

A

uses the contrapositive of a conditional statement to prove the statement itself

46
Q

‌Proof of contradiction

A

It is when we assume a proposition is not true and show that it leads to a contradiction, which proves that the proposition is true.

47
Q

‌Proof of Exhaustion

A

To prove a statement, you must break the statement into a finite amount of cases.

EXAMPLE:
To prove If p then q:
p₁ then q
p₂ then q

48
Q

Proof of Counterexamples

A

Providing a single example where a statement is false to disprove it

49
Q

Mathematical Induction

A

a technique of proving a statement, theorem, or formula which is thought to be true, for each and every natural number n.

50
Q

‌Appeal to the Authority

A

occurs when we accept a claim merely because someone tells us that an authority figure supports that claim.

51
Q

Appeal to Force

A

argumentation using force or the threat of force to convince others to accept an argument’s conclusion

52
Q

Appeal to Ignorance

A

occurs when you argue that your conclusion must be true, because there is no evidence against it

53
Q

Appeal to Pity

A

an attempt to persuade others by exploiting and provoking feelings of guilt or pity instead of presenting factual evidence.

54
Q

Appeal to people

A

argues that a claim is true simply because that’s what a large number of people believe.

55
Q

‌Argumentum Ad Hominem

A

occurs when, instead of addressing someone’s argument or position, you irrelevantly attack the person or some aspect of the person who is making the argument.

56
Q

Circular Argument

A

a logical fallacy wherein it tries to prove itself using its conclusion as evidence.

57
Q

Equivocation

A

happens when a word or phrase in an argument has more than one meaning, and the person switches between those meanings to make their point seem more logical or convincing than it really is.

58
Q

Fallacy of Division

A

This happens when a person believes that what is true for the entire group or thing also holds true for each of its component pieces

59
Q

False Dilemma

A

logical fallacy that presents only two options or sides to an issue when there are actually more complexities.

60
Q

Hasty Generalization

A

happens when someone makes a decision or forms a conclusion without having enough evidence or by only looking at one side of the story.

61
Q

Red Herring

A

occurs when someone introduces an irrelevant point or topic to divert attention from the original issue.

62
Q

Slippery Slope (or snowball/‌domino theory)

A

Also known as Domino Fallacy, It is a logical fallacy that assumes a cause-and-effect relationship between events without evidence.

63
Q

Strawman Fallacy

A

the distortion of someone else’s argument to make it easier to attack or refute.

11 GEN-MATH Q2 (3 decks)

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