Math Truth Tables

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Conditional statements are usually in the form of p %u2183 q or p→q. In this statement, the “if’ statement (p in this case) is referred to as antecedent with the next statement (q in this case) being referred to as consequent (TutorVista, 2010). For the example p %u2183 q, it implies that provided that p is correct, then q is also correct. These statements are usually accompanied by a converse statement. In addition, a statements truth value is usually its falseness or its truth. As illustrated below, all the statements that are meaningful have to be accompanied by a truth value.

Example 1

Consider the following statement which tries to show the effectiveness of a specific shampoo as proven effective and not causing baldness. The statement reads “You either use Kleen; an approved shampoo or you end up bald”. This slogan is very sensitive especially to the people with the worry of losing their hair.

Let p be: You do not use Kleen; a professionally approved shampoo

Then q for this case is: You end up bald.

If q is correct and also p is correct, then “if p hence p” can also be said to be true. That is to say that the above statement has the following conditional statement:  “If you do not use Kleen; a professionally approved shampoo, then you end up bald”. This statement can also be said to be comparable to; you use Kleen; a professionally approved shampoo or you end up bald. Symbolically, this statement can be denoted as; (~p)%u0474q.  Since (~p%u0474q) ≡p→q , we can take a negative (Chandon, 2006) on both sides and end up with;

~(~p%u0474q) ≡~(p→q) 

Incorporating the law of Demorgan, we have;

~(~p) %u028C~q≡ ~ (p→q)

Example 2

Consider a slogan that reads: Rock like Chris with Denim Jeans (Schroyens et al., 2001). The advertisement statement reads: If you wear Denim jeans, you will rock like Chris. This statement can be negated as follows: “You do not wear Denim jeans so you will not rock like Chris”.  For inverse of the conditional statement, we are aware that;

 p→q ≡(~p)%u0474q

The commutative statement is denoted as;  ≡q%u0474 (~p)

≡~q→~p

Therefore, ~q→~p is referred to as the inverse of p→q.

Example 3

The deceptive Goodyear slogan reads as follows: “In all the best tires, there is Goodyear writing on them” (Whitney, 2011). From this slogan, let us evaluate the false and true statements. The conditional statement will read as follows: “If you talk about the best tires, then you have to see a Goodyear writing on them. Then the inverse statement will be; “In case you see tires written Goodyear in them, then they have to be the best tires”. In addition, this slogan will have the converse statement which reads as follows: If tires are not the best, then you will not see a writing of Goodyear on them”. The contrapositive will therefore be; “If you cannot see the good year writing in tires, then they cannot be the best tires”. As it has been noted in this example, it is clear that the conditional statements will come in pairs or even more (Garth, 2002; Schroyens, Walter & Simon, 2003). In that case, we can have either the contrapositive and conditional statement being regarded as true or both sentences being false (Suber, 1997). In case the conditional and contrapositive statements are true, then the inverse statement will not be true (Jonathan et al., 2003).

Example 4

Suppose an advertisement slogan reads as follows: “If you use vision revision term papers, you will excel in the final examination”. This statement can be divided in to two sensible parts:

p: You use the vision revision papers.

q: You excel in the final examination.

When we translate this statement into the form of symbols, the statement can be denoted in the form of p→q. To get the conditional statement, let us consider the four perspectives of the statement illustrated on the truth table:

From the truth table illustrated above, it can be said that the adjacent is true (the statement coming before connective →is correct). In other cases, the conditional statement can be said to be true.

Example 5

Consider the following slogan about Hair Grow: “If you apply Hair Grow twice per day, then you will not go bald”. First, let us connect this simple statement with letters.

p: You apply Hair Grow twice per day.

q: You never become bald.

If this statement is converted into symbolic form, we get; ~p→ (p %u028C q).

In order to determine the truth values for this conditional statement, we have to first determine the truth values for ~p and (p %u028C q) coming after and before the connective (Koehler, 2002). Given this conditional statement, the original statement can be denoted as untrue.

Example 6

Consider the advertisement slogan written; “Crest battles tooth cavity”. This statement can fully be written as; “If you use Crest toothpaste, then you will have no cavity” (Brown, 2006). This statement can explicitly be represented as follows:

p: You use Crest toothpaste.

q: You do not get tooth cavity.

q→p: You do not get cavity, therefore you use Crest toothpaste.

In this slogan, q→p is represented exhaustively in the explicit ideal making the conclusion: “You do not get tooth cavity” to be inferred directly.

Example 7

The following statement was found on an advertisement slogan that was aimed at increasing the sale of hair conditioner.

p: If you use hair conditioner.

q: your hair becomes as healthy as Jeylo’s.

p→q= If you use hair conditioner, your hair becomes as healthy as Jeylo’s. In the converse format, the statement will be; q→p= If my hair is healthy like Jeylo’s, then you use hair conditioner. In the conditional statement, this advertisement claim can be regarded as false although the inverse is found to be true.

Example 8

Consider the advertisement promoting the use of a dishwasher detergent.

p: If an individual uses a dishwasher detergent.

q: The dishes will be flawless.

p→p= If an individual uses a dishwasher soap, the dishes will be flawless: This is the conditional statement. The converse to this conditional statement will be as follows: q→p= If your dishes are flawless, an individual uses liquid dishwasher soap. In the conditional statement, this claim is also noted to be false. The converse statement is however highlighted to be true. That is; if your dishes are flawless, an individual uses liquid dishwasher soap.

Example 9

In an advertisement on the use of Kleenex tissues, the claim is divided into the following sections:

p: Do not inject cold into your body.

q:  Utilize Kleenex tissues.

The conditional statement p→q is; You do not inject cold into your pocket if you use Kleenex tissues. The converse of the conditional statement q→p is; You utilize the Kleenex, You do not inject cold into your body. Given that Kleenex effectiveness is seen through prevention, then the conditional statement for this claim is true. That is; You utilize the Kleenex, you do not inject cold into your body.

Example 10

A slogan on jeep advertisement read as follows: “If it is rated for trails, it is Daimler 4x4 Jeep” (Chandon, 2006).

p: If a vehicle is for trails.

q: It is Daimler 4x4 Jeep.

The converse statement for the conditional statement q→p will be; “If is Daimler 4x4 Jeep, then it is rated for trails”. The converse to the conditional statement is in this case true and based on this understanding, the correct statement will be: ““If it is Daimler 4x4 Jeep, then it is rated for trails”.

Example 11

A slogan on removal of statin in laundry using Shout reads; “You have stains? Then shout them out”.

p: If an individual uses Shout laundry stain remover.

q: The stain will finally fade away.

The conditional statement will read as follows: If an individual uses Shout laundry stain remover, the stain will finally fade away. The converse to this statement will read as follows: If the stain finally fades away, an individual uses Shout laundry stain remover. The conditional statement in this case can be highlighted as true: That is: If an individual uses Shout laundry stain remover, the stain will finally fade away.

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