1B propositions and truth values

Suppose the statement​ 'p or​ q' is true. Of what can you be​ certain? Explain your reasoning.

One or both of the statements are true. An​ 'or' statement is true if either or both propositions are​ true, and false only if both propositions are false.

negation

logical opposite of a proposition, makes the opposite claim of p " Joan is sitting in the chair -> Joan is not sitting in the chair "

exclusive "or"

one or the other but not both ex: I live in state A or state B.

Explain why the contrapositive is called the inverse of the converse. Is the contrapositive also the converse of the​ inverse?

The converse of a conditional statement​ "if p, then​ q" is; "if q, then p." The inverse of a conditional statement​ 'if p, then​ q' is; "if not p, then not q." ​So, the inverse of the converse is "if not q, then not p," which is known as the contrapositive. The converse of the inverse would be "if not q, then not p, "which is the same as the contrapositive.

Explain the meaning of the given​ statement, which contains a multiple negation. Then answer the question that follows. Paul denies that he opposes the plan to build a new dorm. Does Paul support building a new​ dorm?

The first negation​ "opposes the​ plan" makes it seem that Paul would not support the new dorm. But the statement negates this​ negation, "Paul denies that he opposes the​ plan." Therefore, it seems that Paul supports the plan for the new dorm.

If​ elected, I promise to protect Medicare and to expand education funding. Determine the truth of the campaign proposition if the candidate wins the election and expands education funding but does not protect Medicare.

. False, because only one condition of the conjunction is met.

Rephrase the following statement using one or more conditional statements​ (if p, then​ q). Without love​, you​ don't have hate. Without hate​, you have nothing.

If you​ don't have love​, then you​ don't have hate. If you​ don't have hate​, then you have nothing.

The statement​ "Mathematics is​ fun" is what type of​ statement? Explain your reasoning.

It is a proposition because the statement makes a claim that may be either true or false.

Determine if the following statement is a​ proposition, and give an explanation. Back to the future

No, because the statement does not make a​ claim, and it is not a complete sentence.

Which of the following has the form of a conditional​ statement? Explain your reasoning.

The phrase​ 'if x, then​ y' has the form of a conditional statement because conditional statements use the words​ "if" and​ "then."

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). We intend to catch​ him, dead or alive.

The statement makes sense because when the man the statement is referring to is​ caught, he must be either dead or alive.

When are two statements logically​ equivalent?

Two statements logically equivalent if they have the same truth values.

disjunction

a compound statement made with "or" , we assume the "or" is inclusive so the disjunction is true if either or both propositions are true

inclusive "or"

either or both ex: For my next birthday, I would be thrilled to receive a new car or money.

truth value

either true or false

conditional propositions ( if p then q )

propose something to be true ( the "then" part of the statement ) on the condition that something else is true ( the "if" part of the statement )

logically equivalent

when two statements share the same truth values; if one is true, so is the other. if one is false, so is the other.

truth table

table with a row of each possible set of truth values for the propositions being considered

define disjunction

Given two propositions p and​ q, the statement p or q is called their disjunction. It is false only if p and q are both false. " I have a voice or I can sing "

What is a​ proposition?

A proposition is a statement that makes a claim​ (either an assertion or a​ denial). It may be either true or false, and it must have the structure of a complete sentence. "I did not take the pencil" (complete sentence that makes a denial) "the sun is shining" (complete sentence that makes an assertion) "the sky is purple" (complete sentence that makes an assertion)

The statement​ "If it's a​ dog, then it is a​ mammal" may be rephrased as which of the​ following? Explain your reasoning.

Being a mammal is necessary for being a dog. A common alternative way of stating​ "if p, then​ q" is​ "q is necessary for​ p."

Write the following conditional statement in the form​ (a) 'p is sufficient for​ q' and​ (b) 'q is necessary for​ p.' If you​ believe, then you can achieve - Tupac Shakur

Believing is sufficient for achieving. Achieving is necessary for believing.

converse

if q then p if you are breathing then you are sleeping

propositions

building blocks of arguments, makes a claim (assertion or denial) that may be either true or false

Write the​ converse, inverse, and contrapositive of the following proposition. Of these four​ propositions, state which pairs are equivalent. If Jose has a family​, then he has a daughter.

converse: If Jose has a daughter​, then he has a family. inverse: If Jose does not have a family​, then he does not have a daughter. contrapositive: If Jose does not have a daughter​, then he does not have a family. logically equivalent: conditional and contrapositive converse and inverse

Write the​ converse, inverse, and contrapositive of the following proposition. Of these four​ propositions, state which pairs are equivalent. If lightning is flashing​, then it is cold outside.

converse: If it is cold outside​, then lightning is flashing. inverse: If lightning is not flashing​, then it is not cold outside. contrapositive: If it is not cold outside​, then lightning is not flashing. logically equivalent: converse and inverse conditional and contrapositive

contrapositive

if not q then not p if you are not breathing then you are not sleeping

The following propositions have the form p and q. State p and​ q, and give their truth values. Then determine whether the entire proposition is true or​ false, and explain why. bananas are vegetables and peas are fruit

p- bananas are vegetables q- peas are fruit what is the truth value of p? : false what is the truth value of q? : false is the entire proposition true or false? : the entire proposition is false because p is false and q is false

Explain when​ 'p and​ q' is true.

'p and​ q' is true when both p and q are true.

Explain when​ 'p or​ q' is true.

'p or​ q' is true when both p and q are true. ​'p or​ q' is true when p is false and q is true. ​'p or​ q' is true when p is true and q is false.

define conditional

Given two propositions p and​ q, the statement if​ p, then q is called their conditional. It is true in all cases except the case in which p is true and q is false. " If I have a voice, then I can sing "

define conjunction

Given two propositions p and​ q, the statement "p and q" is called their conjunction. It is true only if p and q are both true. " I have a voice and I can sing "

inverse

if not p then not q if you are not sleeping then you are not breathing

conditional

if p then q if you are sleeping then you are breathing

Make a truth table for the given statement. Assume that​ p, q,​ r, and s represent propositions. p or left parenthesis not p right parenthesis p or (not p)

p not p p or (not p) T F T F T T

Explain when​ 'if p, then​ q' is true.

​'if p, then​ q' is true when both p and q are false. ​'if p, then​ q' is true when p is false and q is true. 'if p, then​ q' is true when both p and q are true.