True and false statements
We can use words and symbols to make meaningful sentences, also called statements For example: a) Mary snores. b) A healthy warthog has four legs. c) 2 + 3 = 5. d)x+ 5 = 7. e)
0 sinx dx= 2. f) ∀x∈R ∃y∈Rsuch thaty 2 =x. g) x/x= 1. h)3∈[1,2). i) Dr Pfaff is the president of the United States.
The article discusses various statements that utilize unusual symbols, such as ∀ (for all), ∈ (belongs to), and ∃ (there exists), with ∀ referred to as the universal quantifier and ∃ as the existential quantifier It categorizes these statements into three groups: true statements (b, c, e), false statements (f, h, i), and those that are neither true nor false (a, d, g).
Ambiguity in statements arises from a lack of sufficient information, leading to uncertainty about their truth value For instance, when discussing "Mary," we must clarify which Mary is referenced and whether she is currently snoring or has a history of snoring Similarly, for mathematical statements, we need to confirm values, such as whether x equals 2 or if x is a nonzero number These ambiguous statements, which cannot be definitively categorized as true or false, are prevalent in real-life situations.
A proposition is defined as a statement that can be classified as either true or false, with its truth value represented as T for true and F for false In this context, statements b, c, e, f, h, and i qualify as propositions, while a, d, and g do not However, by modifying statements d and g to d’) ∀x: x + 5 = 7 and g’) ∃x: x/x = 1, they can be transformed into propositions The modified proposition d’ is false, as 1 + 56 does not equal 7, whereas the modified proposition g’ is true since, for instance, if we take x = 2, then 2/2 equals 1.
Logical connectives and truth tables
Understanding key mathematical terms is crucial for effective learning Essential logical terms include "not," "and," "or," "if then," and "if and only if." The term "and" is particularly important as it combines two sentences to create a new, distinct statement For instance, when we combine sentences c and h, we generate a new sentence that differs from the originals.
In logical expressions, the statement "2 + 3 = 5 and 3∈[1,2)" is false because 3 does not belong to the interval [1,2) The meanings of such compound sentences can be easily understood through their individual components Basic logical terms, or connectives, can be viewed as operations on sentences While grammatical conventions dictate proper sentence structure, we will adopt an algebraic approach to writing compound sentences For simplicity, we may place the word "not" at the beginning of a sentence, even if it deviates from traditional grammar rules For instance, instead of stating "Dr Pfaff is not the president of the United States," we might write "not Dr Pfaff is the president of the United States," as both forms serve our analytical purposes.
We will use the following symbolic notation.
Definition 1.1 Thenegationof a statementP is denoted byơP, verbalized as “it is not the case thatP” or just “notP”.
The operation of negation always reverses the truth value of a sentence.
We summarize this in this truth table:
Recall thatT stands for true and F for false.
Definition 1.2 We write P ∧Q for “P and Q”, the conjunction of two sentences This is true precisely when both of the constituent parts are correct, but false otherwise.
This is in line with normal everyday usage of the word “and” Nobody would deny that I am telling the truth if I say
(2 + 2 = 4)∧(3 + 3 = 6), nor would anyone hesitate to call me a liar if I boldly announced that
Here is the truth table for the conjunctionP∧Q:
Definition 1.3 We writeP∨Qfor “P or Q”, the disjunction of two state- ments.
In mathematics, the term "or" differs from its common usage in everyday language, where it typically implies an exclusive choice between two options, as seen in restaurant menus that offer either soup or salad However, in mathematical contexts, "or" is used inclusively, meaning "at least one, possibly both." This distinction is crucial for understanding mathematical statements and their implications.
(2 + 2 = 4)∨(3 + 3 = 6) are true The only “or” statement that is false is the one with both component parts false, as for example
In everyday conversation, saying "I will take you to the movies or buy you a candy bar" implies that fulfilling either promise would be satisfactory, and doing both would not be considered deceitful This scenario exemplifies the correct mathematical interpretation of "or," which is further explained by the truth table of disjunction.
The usage of the word "or" in mathematics may seem arbitrary at first glance, but it is a widely accepted interpretation among mathematicians in textbooks and journal articles This specific application of disjunction fosters meaningful relationships between logical connectives, similar to the fundamental identities and laws of algebra that many are familiar with from high school.
To be fair, we mention that some people work with the logical operation exclusive or, denoted as∨or⊕in a truth table:
Notice that in this case, P ⊕Q is true precisely when only one of the components is true and the other is false.
Definition 1.4 A sentence of the form “ifP thenQ” is written symbolically
A conditional statement, denoted as P ⇒ Q or P → Q, expresses that P implies Q This can be interpreted in various ways, such as "P only if Q," indicating that P is sufficient for Q, while Q is necessary for P In this context, P is referred to as the hypothesis or antecedent, and Q is known as the conclusion or consequent.
An "if then" statement is clearly defined in logical contexts, distinguishing it from casual usage in everyday conversation It is crucial to recognize that any two statements can be combined to create a conditional, with each part capable of being true or false.
A conditional statement is considered false when the antecedent is true and the consequent is false; in all other cases, the conditional is deemed true Therefore, a conditional is regarded as true whenever the statement that follows is accurate.
“if” and preceding “then” is false As this convention may shock your tender sensibilities, we will try to motivate our reasons for choosing it by relating to some examples.
A conditional statement, represented as P ⇒ Q, can be understood as a promise dependent on a specific condition being met For instance, if I promise you an A in the class if your average is 90% or higher, you would rightfully expect an A with an average of 92.3% However, if your average is only 90%, you would be disappointed if you received a B instead Conversely, if your average falls below 90%, such as 89.7%, I am free to assign any grade without violating my promise, as the original statement does not address outcomes below the 90% threshold In this context, P signifies “your average is 90% or better,” while Q denotes “your grade is A,” illustrating the conditional nature of the promise.
1 Your average is92.3%and your grade is A.
2 Your average is92.3%and your grade is B.
3 Your average is89.7%and your grade is A.
4 Your average is89.7%and your grade is B.
In the context of my promise, the only clear instance of dishonesty stems from situation number 2, as I have upheld my commitments in the other three cases If you fail to meet the requirements, such as achieving an average of 89.7%, you might accept receiving a B However, my decision to grant you an A out of generosity does not make me a liar.
Comedians have known about and used the mathematical interpretation of a conditional statement for a long time Here it is:
“If you had two million dollars, would you give me one million?”
“If you had two thousand dollars, would you give me one thousand?”
“If you had twenty dollars, would you give me ten?”
The essence of conditional statements is that one can make promises without deceit, as long as the condition remains unfulfilled It's crucial to understand that the implication P ⇒ Q does not ensure the truth of either P or Q on their own Instead, the validity of a conditional statement highlights the relationship between a hypothesis and its conclusion Therefore, even if the conditional holds true, the accuracy of Q can only be confirmed after establishing the truth of P.
Here is another example to illustrate that a false statement implies any- thing: let’s prove both
(1) if2 + 2 = 5, then3 = 0, and (2) if2 + 2 = 5, then3 = 3.
We begin with the premise that if 2 + 2 equals 5, then it follows that 4 equals 5 By subtracting 5 from both sides of the equation, we arrive at the conclusion that -1 equals 0 This illustrates the application of valid algebraic principles to demonstrate a logical inconsistency.
−3, we get 3 = 0 What do you think? Did we prove that 3 = 0? Of course not We proved it from a false assumption The entire statement (1) must be regarded as true.
For (2), we have the hypothesis2+2 = 5 Multiply both sides by0and add
3to both sides We obtain a valid result3 = 3from an erroneous assumption.
To summarize, the truth table for the conditionalP ⇒Qis as follows:
In logic, the notation P ⇔ Q signifies that "P if and only if Q," which is referred to as a biconditional or equivalence This relationship indicates that P is both necessary and sufficient for Q Additionally, some may represent this concept using the symbol P ↔ Q The phrase "if and only if" is commonly abbreviated to "iff."
A biconditional statement expresses that two surrounding statements convey the same truth value, meaning both are either true or false It's important to note that simply stating a biconditional does not guarantee its truth There are no predefined limitations on the types of sentences that can be connected by the biconditional symbol (⇔), and the relationship between the components may not be obvious The truth or falsity of the entire biconditional is determined exclusively by analyzing the individual statements involved.
The statement "(2 + 2 = 4) if and only if (7 divides 1001)" is considered true, even if the connection between the two parts seems unclear In logical terms, when two true propositions are linked by "if and only if," the overall statement holds validity What is your assessment of this sentence? Is it true or false?
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