During the learning engagement, the learners should be able to:

1. Explain the importance of utilizing mathematical language and its conventions;

2. Differentiate the corresponding mathematical and English languages.

3. Utilize logical connectives in expressing certain mathematical concept;

4. Construct a truth table.

This lesson will introduce you to a mathematical way of thinking that can serve you in a wide variety of situations. Often when you start working on a mathematical problem, you may have only a vague sense of how to proceed. You may begin by looking at examples, drawing pictures, playing around with symbols, re-read the problem to focus on more of its details, and so forth. The closer you get to a solution, however, the more your thinking has to crystallize. And the more you need to understand, the more you need language that expresses mathematical ideas clearly, precisely, and unambiguously.

This lesson aims to give you some ideas about the characteristics and conventions of mathematics as a language that is a foundation for dealing with everyday life. It also focuses on logical symbols, logical connectives, truth tables, tautologies, and self-contradiction.

Self Activity 1: Left or Right?

Consider the following conversation of Mr. Pandemic and Mrs. Covid. Answer the questions that follow. No need to submit the answers. God bless you!

Guide Questions:

1. What can you say about the conversation?

2. Can you give another example with the same idea with that of the given conversation?

Self Activity 2: Where Do I Belong?

Instruction: Classify each phrase, according to the following categories as indicated on the table. Write your answer on the table. No need to submit the answers. God bless you!

B. Translate each symbolic notation into a sentence. Let p represents “Today is Monday” and q represents “The weather is good.”

C. Answer the following questions:

1. From your answers on Test A, what are your observations? How did it help you?

2. From your answers on Test B, what are your observations? How did it help you?

Formal Discussion of Concepts

Definition of a Mathematical Language

For thousand years, mathematicians had developed spoken and written natural languages that are highly effective for expressing mathematical language. This mathematical language has developed and provides a highly efficient and powerful tool for mathematical expression, exploration, reconstruction after exploration, and communication. Its power comes from simultaneously being precise and yet concise. But mathematical language is being used poorly because of poor understanding of the language. The mathematical language and logical reasoning using that language form the everyday working experience of mathematics.

The mathematical language is the system used to communicate mathematical ideas. This language consists of some natural language using technical terms (mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas. The mathematical notation used for formulas has its own grammar and shared by mathematicians anywhere in the globe.

Characteristics of a Mathematical Language

The first characteristic of a mathematical language is being precise. Precision in mathematics is like a culture of being correct all the time. Definition and limits should be distinction. Mathematical ideas are being developed informally and being done more formally, with necessary and sufficient conditions stated up front and restricting the discussion to a particular class of objects. Mathematical culture of precision has developed a precise, highly symbolic language, and a dialect manner that allows for the adaptation, adjustment and cumulative refinement of concepts based on experiences, and mathematical reasoning is expected to be correct.

The second characteristic of a mathematical language is being concise or shows simplicity. Being concise is a strong part of the culture in mathematical language. The mathematician desires the simplest possible single exposition at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the highest possible level.

The third characteristics of a mathematical language is being powerful. It is a way of expressing complex thoughts with relative ease. The abstraction in mathematics is the desire to unify diverse instances under a single conceptual framework and allows easier penetration of the subject and the development of more powerful methods.

According to Galileo Galilei, “Mathematics is the language in which God has written the universe”. It can be attributed that mathematics is a universal language because the principles and foundations of mathematics are the same everywhere around the world.

Conventions of Mathematical Language

Mathematical languages have conventions and it helps individual distinguish between different types of mathematical expressions. A mathematical convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians. For example, one evaluates multiplication before addition following the principle of PEMDAS (Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction). Mathematicians abide by conventions to be able to understand what they write without constantly having to redefine basic terms. Almost all mathematical names and symbols are conventional.

Any idea, no matter how simple it is, would become very difficult if there is no knowledge of the language in which the ideas were presented. Students have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language—the language of mathematics.

Mathematical Expressions, Sentences, and Statements

A mathematical expression is the mathematical analogue of an English noun. It is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does not state a complete thought; in particular, it does not make sense to ask if an expression is true or false. A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. In a mathematical sentence, it makes sense to ask about the truth of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false?

Mathematics is a universal language. It is with own set of vocabulary, rules, and system of communication which requires more than just knowing what those words and sentences are. What do you think is the verb counterpart in Mathematical Language? Consider the mathematical sentence 3 +4 = 7. The verb is ‘ = ’. If you read the sentence as ‘three plus four is equal to seven’, then it’s easy to ‘hear’ the verb. Indeed, the equal sign ‘ = ’ is one of the most popular mathematical verbs. Numbers have lots of different names: 5, 2+3, 102, (6-2)+1. Just like an English word may have different synonyms: enemy, adversary, foe, opponent, etc

Sentences can be true or false. The notion of truth (i.e., the property of being true or false) is of fundamental importance in the mathematical language. Languages have conventions. In English, for example, it is conventional to capitalize proper names (like ‘Carol’ and ‘Christmas Day’). This convention makes it easy for a reader to distinguish between a common noun (like ‘carol’, a Christmas song) and a proper noun (like ‘Carol’). Mathematics also has its conventions. There are rules on how to attach prefixes and suffixes.

A statement is a declarative sentence that is either true or false, but not both true and false. A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas.

Mathematical Symbols

The truth value of a simple statement is either true (T) or false (F). The truth value of a compound statement depends on the truth values of its simple statements and its connectives. A truth table is a table that shows the truth value of a compound statement for all possible truth values of its simple statements.

1. A statement like “ 7 < 10” is true.

2. A statement like “A cat is a fish” is false.

3. A statement like “ x < 5 ” is true for some values of and false for others.

Examples of statements include the following:

1. Taft is a barangay of Surigao City. (True)

2. Metro Manila is the largest city of the Philippines in population. (False)

3. 2 + 3 = 5. (True)

4. 3 < 0. (False)

The following are not statements:

1. Surigao City is the best city. (Subjective)

2. Help! (An exclamation)

3. Where were you? (A question)

4. The rain in Cabadbaran. (Not a sentence)

5. This sentence is false. (Neither true nor false!)

Logical Connectives

If two or more statements are joined, or connected, then we can form compound statements. These compound statements are joined by logical connectives “and”, “or”, “if then”, and “if and only if”.

Symbolic form of compound statements with p and q as simple statements.

Statements are represented symbolically by lowercase letters (e.g., p, q, r, and s) and new statements can be created from existing statements in many ways.

4. IF THEN (→ )

The statement “If p, then q”, denoted by p → q (read as "If p implies q"), is called an implication or conditional statement; p is called the hypothesis, and q is called the conclusion.

Example. Consider again the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.

· The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.

· The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.

· The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is true, and is equivalent to the original conditional.

In the next truth table, it displays the various truth values for these four conditionals.

Notice that the columns of truth values under the conditional p→q and its contrapositive are the same. When this is the case, we say that the two statements are logically equivalent. In general, two statements are logically equivalent when they have the same truth tables. Similarly, the converse of p→q and the inverse of p→q has the same truth table; hence, they, too, are logically equivalent.

• A conditional statement and its contrapositive are logically equivalent. The converse and inverse of a conditional statement are logically equivalent.

Negation of a Conditional

The negation of a conditional statement is logically equivalent to a conjunction of the antecedent and the negation of the consequent.

~(p → q) is is equivalent to p ^ ~q

5. IF AND ONLY IF (↔ )

The connective “p if and only if q”, called a biconditional and written p ↔ q, is the conjunction of p → q, and its converse q → p. That is, p ↔ q is logically equivalent to (p → q)^(q → p). The truth table of follows.

Notice that the biconditional p ↔ q is true when p and q have the same truth values and false otherwise. Often in mathematics the words necessary and sufficient are used to describe conditionals and biconditionals. For example, the statement “Water is necessary for the formation of ice” means “If there is ice, then there is water.” Similarly, the statement “A rectangle with two adjacent sides the same length is a sufficient condition to determine a square” means “If a rectangle has two adjacent sides the same length, then it is a square”. Symbolically we have the following:

p → q means q is necessary for p

p → q means p is sufficient for q

p ↔ q means p is necessary and sufficient for q

Example 1. Write the negation of each of the following statements.

a. Some airports are open.

b. All movies are worth the price of admission.

c. No odd numbers are divisible by 2.

Solution:

a. No airports are open.

b. Some movies are not worth the price of admission.

c. Some odd numbers are divisible by 2.

Example 2. Write the negation of the following statements.

a. All bears are brown.

b. No smartphones are expensive.

c. Some vegetables are not green.

Compound Statements and Grouping Symbols

If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. Table above illustrates the use of parentheses to indicate groupings for some statements in symbolic form.

If a compound statement is written as an English sentence, then a comma is used to indicate which simple statements are grouped together. Statements on the same side of a comma are grouped together. See Table below.

TAUTOLOGIES AND SELF-CONTRADICTION

A tautology is a statement that is always true. A self-contradiction is a statement that is always false.

Self Activity 4: Construct Me!

Instruction: Construct a truth table of [(p ^ q) v ~p] → [(p v q) ^~q].

Follow-up questions:

1. What can you say about your final results of the truth table?

2. How will you apply this concept of activity in real-life setting? What is its value to you as a student?