What are equivalent statements?
What are equivalent statements?
Equivalent Statements are statements that are written differently, but hold the same logical equivalence. Case 1: “ If p then q ” has three equivalent statements.
What is the equivalent of a conditional statement?
A conditional statement is logically equivalent to its contrapositive. Converse: Suppose a conditional statement of the form “If p then q” is given. The converse is “If q then p.” Symbolically, the converse of p q is q p.
What is equivalent statement logic?
Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions. In this case, we write X≡Y and say that X and Y are logically equivalent.
How do you know if a statement is logically equivalent?
Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables. p q and q p have the same truth values, so they are logically equivalent.
How do you prove equivalent statements?
An if-and-only-if theorem of form P⇔Q asserts that P and Q are either both true or both false, that is, that P and Q are equivalent. To prove P⇔Q we prove P⇒Q followed by P⇒Q, essentially making a “cycle” of implications from P to Q and back to P….
What pairs of propositions are logically equivalent?
The propositions are equal or logically equivalent if they always have the same truth value. That is, p and q are logically equivalent if p is true whenever q is true, and vice versa, and if p is false whenever q is false, and vice versa. If p and q are logically equivalent, we write p = q….
What does P and Q mean in logic?
3. Conditional Propositions. A proposition of the form “if p then q” or “p implies q”, represented “p → q” is called a conditional proposition. For instance: “if John is from Chicago then John is from Illinois”. The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or consequent.
What is logically equivalent to P and Q?
A compound proposition that is always True is called a tautology. Two propositions p and q are logically equivalent if their truth tables are the same. Namely, p and q are logically equivalent if p ↔ q is a tautology. If p and q are logically equivalent, we write p ≡ q.
What is the truth table of p λ Q → P?
So because we don’t have statements on either side of the “and” symbol that are both true, the statment ~p∧q is false. So ~p∧q=F. Now that we know the truth value of everything in the parintheses (~p∧q), we can join this statement with ∨p to give us the final statement (~p∧q)∨p….Truth Tables.
| p | q | p→q |
|---|---|---|
| T | F | F |
| F | T | T |
| F | F | T |
What is the truth value of P ∨ Q?
The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The truth value of p ∨ q is false if both p and q are false. Otherwise, it is true.
What does P ∧ Q mean?
P ∧ Q means P and Q. P ∨ Q means P or Q. An argument is valid if the following conditional holds: If all the premises are true, the conclusion must be true. So, when you attempt to write a valid argument, you should try to write out what the logical structure of the argument is by symbolizing it.
What is the converse of P → Q?
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S.
Can the converse be true?
If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true….Example 1:
| Statement | If two angles are congruent, then they have the same measure. |
|---|---|
| Converse | If two angles have the same measure, then they are congruent. |
What is the negation of P -> Q?
The negation of p ∧ q asserts “it is not the case that p and q are both true”. Thus, ¬(p ∧ q) is true exactly when one or both of p and q is false, that is, when ¬p ∨ ¬q is true. Similarly, ¬(p ∨ q) can be seen to the same as ¬p ∧ ¬q.
What is a negation example?
When you want to express the opposite meaning of a particular word or sentence, you can do it by inserting a negation. Negations are words like no, not, and never. If you wanted to express the opposite of I am here, for example, you could say I am not here.
How do you do negation?
One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true). Let’s take a look at some of the most common negations….Summary.
| Statement | Negation |
|---|---|
| “There exists x such that A(x)” | “For every x, not A(x)” |
How do you negate implications?
Negation of an Implication. The negation of an implication is a conjunction: ¬(P→Q) is logically equivalent to P∧¬Q. ¬ ( P → Q ) is logically equivalent to P ∧ ¬ Q .
How do you prove an implication?
Direct Proof
- You prove the implication p –> q by assuming p is true and using your background knowledge and the rules of logic to prove q is true.
- The assumption “p is true” is the first link in a logical chain of statements, each implying its successor, that ends in “q is true”.
What is an example of an implication?
The definition of implication is something that is inferred. An example of implication is the policeman connecting a person to a crime even though there is no evidence. An implicating or being implicated.
Can you negate a quantifier?
To negate a sequence of nested quantifiers, you flip each quantifier in the sequence and then negate the predicate. So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : P(x, y) and So the negation of ∃x ∀y : P(x, y) and ∀x ∃y : P(x, y). Again, after some thought, this make sense intuitively.
When can a Biconditional statement be true?
When we combine two conditional statements this way, we have a biconditional. Definition: A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow .
What is the negation of if and only if?
Negation of a Conditional By definition, p → q is false if, and only if, its hypothesis, p, is true and its conclusion, q, is false. It follows that the negation of “If p then q” is logically equivalent to “p and not q.”
What is the difference between if and if and only if?
To say “A if and only if B” means that A is true if B is true, and B is true if A is true. That is, A is necessary and sufficient for B. Succinctly, A only if B is the logic statement A⇒B.
Does if and only if go both ways?
IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. So it is essentially and “IF” statement that works both ways. Note that IF AND ONLY IF is different than simply ONLY IF….
What is the symbol of if and only if?
Basic logic symbols
| Symbol | Name | Read as |
|---|---|---|
| ⇔ ≡ ↔ | material equivalence | if and only if; iff; means the same as |
| ¬ ˜ ! | negation | not |
| Domain of discourse | Domain of predicate | |
| ∧ · & | logical conjunction | and |
How do you do an if and only proof?
To prove a theorem of the form A IF AND ONLY IF B, you first prove IF A THEN B, then you prove IF B THEN A, and that’s enough to complete the proof. Using this technique, you can use IF… THEN proofs as well as IF AND ONLY IF proofs in your own proof.
What does necessary and sufficient mean in logic?
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. The assertion that a statement is a “necessary and sufficient” condition of another means that the former statement is true if and only if the latter is true.