Figuring out the best way to study mathematics can be overwhelming and difficult. Luckily, we have composed a very helpful series about Math Symbols and Their Meanings so you can understand the language behind mathematics easier and faster.
Logic is the unlocking key to understanding and operating in mathematical language. Moreover, in exercises and problems that require proof or a demonstration, students are required to solve problems with a logical method. In our previous article, we have discussed set theory which is a subarea of basic mathematical logic.
Components Of Mathematical Logic
Mathematical statements, also referred to as propositions are well-defined affirmations that can be either true or false, such as:
- 146 – 31 = 115 … True statement
- 1 x 0 = 1 …. False statement
Propositional Logic
A mathematical system that explains the propositions/statements and how they relate to each other, also known as Boolean Logic. Statements are composed of variables and logical connectives. In general, variables are indicated by lower-case letters, such as a, b, p, x.
Logical connectives are indicated by:
“¬”
This symbol defines a logical negation. Given the variable p, “¬p” (non-p) is false only if “p” is true.
Example: Given the statement p: 6 + 7 = 12 (false). The logical negation: “It is not the case that p is true”, then “¬p” is true because p is false.
“∧”
This symbol defines logical conjunction (and). Given the variables p and q, the proposition “p ∧ q” is true only if both variables are true.
Example: Given the statements p and q, where p is 5 + 5 = 10, q is 4 + 4 = 8, the following statement is a logical conjunction: “p ∧ q” is True. However, if q is false, as in 4 + 4 = 6, the following statement: ““p ∧ q” is False.
“∨”
A logical disjunction, named “or”, in which the statement “p∨q” is true if variable p or q is true. When it comes to disjunctions, one of the statements must be true, they cannot be both false.
Example: Given the statements p and q, where p is 5 + 5 = 10 (true), q is 4 + 4 = 7 (false), the following statement is a logical disjunction: “p ∨ q” is True.
However, if p is 5 + 5 = 11 or q is 4 + 4 = 7, this is False, as both statements are false. Therefore, when it comes to logical disjunctions, at least one statement must be true.
“→”
This symbol indicates a relation of implication between two variables. Given the statement “p→q”, it means that if p is true, then q is true, as well. Things get a bit confusing when p is false, as there is no relevance of the value of statement q.
Example: Given the following statements, p is 3 = 3, and q is 32 = 6, then “p→q” is True; p is true which implies that q is also true.
When p is false, it implies that q is also false (regardless of the fact that q can actually be true).
“↔”
This symbol indicates that there is a biconditionality between two variables. Therefore, the statement “p↔q” is true when the two are equal in value, thus, both are true or both are false, whereas in the implication statement p can be true and q can be false.
Example: Given the statement p: y – 4 = 2, and q: y = 6, then the biconditional relation “p↔q” indicates that: “y – 4 = 2 only when y = 6”.
What Are The Symbols For A True Or False Statement?
“True” and “false” are not logical connectives per se, they are more like values of variables, but in mathematics, they are still used in the area of the connective.
“⊤” – this indicates that a value is always true;
“⊥” – this indicates that a value is always false.
Predicate Logic
Every concept in mathematics has limitations in proving a variable in a statement or the value of a variable. Where propositional logic ends, predicate logic begins. Predicate logic deals with quantifiers. In propositional logic, the only possible statement relations are non, and, or (negation, conjunction, and disjunction). Predicate logic adds to these relations the symbols that have been presented in our previous article about the comparison.
You might have noticed a similarity of linguistic terms – predicates also exist in mathematical language. A statement has two parts. For example, the statement: “X > 7”. X is the subject of the statement. “>7” is the predicate of the statement.
In order to indicate how a predicate behaves in regards to the subject, in mathematics we use quantifiers.
Universal Quantification
When a statement indicates that a condition is met by all the values of a variable, we use universal quantifiers. The symbol for a universal quantifier is “∀”. A statement is indicated as ∀xP(x). ∀xP(x) is true only if P(x) is true, regardless of the value of x.
Example: For any x, x2≥ 0
This statement is true and it would be noted as ∀x ∈ p (x2≥ 0)
A false statement would be: For any X, x2< 0. This can be proven by a simple calculus, however universal quantifiers refer to those aspects of a statement that are true, hence all the values of a variable that exist by the given condition.
Existential quantification
The symbol of existential qualification is “∃”. If in the case of universal quantification, all values had to meet the given condition that makes the statement true, in existential qualification, a statement is true if at least one value meets the condition that makes the statement true.
Example: Determine if the following statement is true, ∃x ∈ p (x > 2).
Solution: The statement is true because x = 3.
In order to determine if a statement is true in the case of existential qualification, we need to find only one value or at least one value that respects the condition ( x > 2). In our case, if x = 3, then x > 2 which proves that the statement is true.
Practice Basic Logic
The easiest way to practice basic mathematical logic and also have a better understanding on how propositional operations work is to create truth tables. This is a great exercise to improve critical thinking and develop a logical sense. A fun fact about basic mathematical logic is that basic logic is the foundation of puzzles and riddles.
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