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yan [13]
3 years ago
5

Consider a relation on the set of all states in the United States given by: two states are related if they have a border in comm

on. Is it an equivalence relation
Mathematics
1 answer:
77julia77 [94]3 years ago
7 0

Answer:

Yes it is an equivalence relation

Explanation:

An equivalence relation is a binary relation between two values that are symmetric, transitive and reflexive. In other words, when we say a value x is equal(using "=") to a value y, there is an equivalence relation between them.

Example, given set {x, y, z} where ~ means equivalence:

x ~ y if y ~ z means symmetric

since x ~ y and y ~ z, then x ~ z means transitive

x ~ x means reflexive

Equivalence relations share a common attribute or attributes(example, a satisfying condition)

The above condition that two states are related from the set of all US states if they have a border in common satisfies the condition of equivalence listed hence it is an equivalence relation.

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For each of the following, use logic laws to decide whether the statement is a tautology contradiction, or neither. a (a) (A B)
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Answer:

The statement (A\rightarrow \lnot B)\land (B\rightarrow A) is a contingency.

The statement (P\rightarrow \lnot P)\land P is a contradiction.

Step-by-step explanation:

A tautology is a proposition that is always true.

A contradiction is a proposition that is always false.

A contingency is a proposition that is neither a tautology nor a contradiction.

a) To classify the statement (A\rightarrow \lnot B)\land (B\rightarrow A), you need to use the logic laws as follows:

(A\rightarrow \lnot B)\land (B\rightarrow A) \equiv

\equiv (\lnot A \lor\lnot B)\land(\lnot B \lor A) by the logical equivalence involving conditional statement.

\equiv (\lnot B\lor \lnot A )\land(\lnot B \lor A) by the Commutative law.

\equiv \lnot B \lor (\lnot A \land A) by Distributive law.

\equiv \lnot B \lor (A \land \lnot A) by the Commutative law.

\equiv \lnot B \lor F by the Negation law.

Therefore the statement (A\rightarrow \lnot B)\land (B\rightarrow A) is a contingency.

b) To classify the statement (P\rightarrow \lnot P)\land P, you need to use the logic laws as follows:

(P\rightarrow \lnot P)\land P \equiv

\equiv (\lnot P \lor \lnot P)\land P by the logical equivalence involving conditional statement.

\equiv P \land (\lnot P \lor \lnot P) by the Commutative law.

\equiv (P \land \lnot P) \lor (P \land \lnot P) by Distributive law.

\equiv F \lor F \equiv F by the Negation law.

Therefore the statement (P\rightarrow \lnot P)\land P is a contradiction.

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