[DEFINE] LOGIC: Terminologies

Question 1

‌Logic

Answer 1

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

Question 2

Proposition

Answer 2

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

Question 3

Conjunction

Answer 3

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

Question 4

Disjunction

Answer 4

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

Question 5

‌Negation

Answer 5

“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.

Question 6

‌Conditional (or implication)

Answer 6

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

Question 7

‌Biconditional

Answer 7

True when both have the same truth value.

Question 8

‌Exclusive-or

Answer 8

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

Question 9

‌Truth Values of proposition

Answer 9

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).

Question 10

Conditional Propositions

Answer 10

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

Question 11

Converse of p-q

Answer 11

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.

Question 12

Inverse of p-q

Answer 12

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.

Question 13

Contrapositive

Answer 13

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

Question 14

Tautology

Answer 14

a proposition that is always true

Question 15

Contradiction

Answer 15

a proposition that is always false

Question 16

Contingency

Answer 16

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

Question 17

‌Proof

Answer 17

a logical argument that shows a mathematical statement is true.

Question 18

‌Premises

Answer 18

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

Question 19

Conclusion

Answer 19

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

Question 20

Number Theory

Answer 20

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

Question 21

Counting

Answer 21

a subset of whole numbers that do not include zero

Question 22

‌Whole

Answer 22

a subset of whole numbers that includes the number zero

Question 23

‌Integers

Answer 23

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

Question 24

Rational Numbers

Answer 24

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

Question 25

‌Irrational Numbers

Answer 25

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

Question 26

Real Numbers

Answer 26

are numbers that include **rational** and **irrational** numbers.

Question 27

‌Imaginary numbers

Answer 27

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

Question 28

‌Complex numbers

Answer 28

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

Question 29

Odd numbers

Answer 29

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

Question 30

Even Numbers

Answer 30

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

Question 31

Prime numbers

Answer 31

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

Question 32

Composite Numbers

Answer 32

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

Question 33

Rule of Inference

Answer 33

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

Question 34

Conjecture

Answer 34

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

Question 35

Axiom (or postulate)

Answer 35

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

Question 36

‌Theorem

Answer 36

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

Question 37

‌Lemma (or pre-theorem)

Answer 37

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

Question 38

‌Corollary (or post-theorem)

Answer 38

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

Question 39

‌Proof

Answer 39

a logical argument that **shows** a mathematical statement is **true**.

Question 40

Divisibility

Answer 40

this means a number can be divided without **leaving a remainder**

Question 41

Greatest Common Divisor

Answer 41

is the **greatest value** that can divide two integers **evenly**.

Question 42

‌Least Common Multiple

Answer 42

the **smallest positive multiple** that two or more numbers share

Question 43

Relatively Prime Integers

Answer 43

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

Question 44

‌Direct Proof

Answer 44

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

Question 45

Proof of Contrapositive (or proof of transposition)

Answer 45

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

Question 46

‌Proof of contradiction

Answer 46

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.

Question 47

‌Proof of Exhaustion

Answer 47

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

Question 48

Proof of Counterexamples

Answer 48

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

Question 49

Mathematical Induction

Answer 49

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

Question 50

‌Appeal to the Authority

Answer 50

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

Question 51

Appeal to Force

Answer 51

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

Question 52

Appeal to Ignorance

Answer 52

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

Question 53

Appeal to Pity

Answer 53

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

Question 54

Appeal to people

Answer 54

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

Question 55

‌Argumentum Ad Hominem

Answer 55

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.

Question 56

Circular Argument

Answer 56

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

Question 57

Equivocation

Answer 57

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.

Question 58

Fallacy of Division

Answer 58

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

Question 59

False Dilemma

Answer 59

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

Question 60

Hasty Generalization

Answer 60

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

Question 61

Red Herring

Answer 61

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

Question 62

Slippery Slope (or snowball/‌domino theory)

Answer 62

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

Question 63

Strawman Fallacy

Answer 63

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