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👩‍🎤 | Nogizaka46 Manatsu Akimoto, Asuka Saito, Mizuki Yamashita, smiled at the hourly wage increase! "Baitoru" New TV-C ...


Photo Nogizaka46 Manatsu Akimoto, Asuka Saito, Mizuki Yamashita, smiled at the hourly wage increase! "Baitoru" new TV-CM broadcast decision

Nogizaka46 Manatsu Akimoto, Asuka Saito, Mizuki Yamashita, smiled at the hourly wage increase! "Baitoru" New TV-C ...

 
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After hearing Saito's lines and judging that the finish was sufficient, the director said, "The production without a test!
 

Nogizaka46's Manatsu Akimoto, Asuka Saito, Mizuki Yamashita and DAIGO will appear in the new TV commercial for "Baitoru" "Sales-san, Ko ... → Continue reading

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Equivalent

Equivalent(How) orEquivalent(Toka) is twopropositionTogethertrueOr togetherfalseWill be true whenLogical operation. EnglishThen equivalence (EQ). Abbreviation for "if and only if"iff Also called.Negative exclusive OR (XNOR) be equivalent to. operatorSymbols such as ⇔, ↔, ≡, =, and EQ are used.

Truth table

proposition Pproposition QP ⇔ Q
truetruetrue
truefalsefalse
falsetruefalse
falsefalsetrue

nature

Basic properties

The basic properties of equivalence are as follows.
( TheLogical conditional(If), TheLogical AND(And))

  • Reflection:
  • Symmetry:
  • Transitional law:

Other

In addition, it has the following properties.
( Thedenial, TheExclusive OR)

  • Antisymmetric law:

Necessary and sufficient conditions

Two conditions p ,q Against p Everything that meets q When you say, "I also meet." p The q To beSufficient conditionsIs "or" q The p To beRequirementsIs. "

Also," p The q It is a sufficient condition to beq The p "It is a sufficient condition to be", " p The q To beNecessary and sufficient conditionsIs "or" p と q AndEquivalentIs. "

Example 1

A number is XNUMXmultipleIn order to be, that number is at leastevenMust be.That is, being an even number is a necessary condition for being a multiple of XNUMX.However, even if it is an even number, it is not always a multiple of XNUMX.

Also, in order for a number to be a multiple of XNUMX, it is sufficient if the number is a multiple of XNUMX.That is, being a multiple of XNUMX is a sufficient condition for being a multiple of XNUMX.However, even if the number is not a multiple of XNUMX, it is not necessarily a multiple of XNUMX.

On the other hand, in order for a number to be a multiple of XNUMX, the number must be at least an even number.That is, being an even number is a necessary condition for being a multiple of XNUMX.If the number is an even number, the number is always a multiple of XNUMX.That is, being an even number is a sufficient condition for being a multiple of XNUMX.That is, being an even number is a necessary and sufficient condition for being a multiple of XNUMX, and both are equivalent values.

Example 2

Natural number variable n Conditions for p(n), q(n) Is defined as follows.

  • p(n): n > 10
  • q(n): 2n > 20

then,p(n) Is q(n) Is a necessary and sufficient condition.That is,n > 10 is 2n It is a necessary and sufficient condition for> 20.

Example 3

Real variable x Conditions for p(x), q(x) Is defined as follows.

  • p(x): x > 0
  • q(x): x2 > 0

then,p(x) Is q(x) Is a sufficient condition.But -1 is q(x) Satisfies p(x) Is not satisfied, so "q(x) Are all real numbers p(x) Is not satisfied.Therefore,q(x) Is p(x) Is not a sufficient condition.Therefore,p(x) Is q(x) Is not a necessary and sufficient condition.

Example 4

Propositional variables with ¬ and ⇔ as logical operations A ,B Conditions for p(A, B), q(A, B) Is defined as follows. (¬ is one mapping from the set {true, false} to the set {true, false}. ⇔ is one mapping from {true, false} × {true, false} to {true, false}. ..A ,B Is the original variable of {true, false}. )

  • p(A, B): ï¿¢ ( A ⇔B ) = True
  • q(A, B): (ï¿¢A ) ⇔B = True

then,p(A, B) Is q(A, B) Is a necessary and sufficient condition.That is, "¬ ( A ⇔B ) = True "is" (¬¬A ) ⇔B = True ”is a necessary and sufficient condition.

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