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Mệnh đề đảo. Mệnh đề tương đương

Mệnh đề (Q Rightarrow P)được gọi là mệnh đề đảo của mệnh đề (P Rightarrow Q). Nếu cả hai mệnh đề (P Rightarrow Q) và (Q Rightarrow P) đều đúng thì ta nói P và Q là hai mệnh đề tương đương. Kí hiệu là (P Leftrightarrow Q).
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1. Lý thuyết

+ Định nghĩa: Mệnh đề \(Q \Rightarrow P\)được gọi là mệnh đề đảo của mệnh đề \(P \Rightarrow Q\).

+  Chú ý: Mệnh đề đảo của một mệnh đề đúng không nhấ🐬t th💮iết là đúng.

Ví dụ: “Nếu \(a = 2\) thì \({a^2} - 4 = 0\)” là mệnh đề đúng. “Nếu \({a^2} - 4 = 0\) thì \(a = 2\)” là mệnh đề sai

+ Nếu cả hai mệnh đề \(P \Rightarrow Q\) và \(Q \Rightarrow P\) đều đúng thì ta nói P và Q là hai mệnh đề tương đương. Kí hiệu là \(P \Leftrightarrow Q\).

+ Các cách phát biểu mệnh đề \(P \Leftrightarrow Q\):

  • “P tương đương Q”
  • “P khi và chỉ khi Q”
  • “P nếu và chỉ nếu Q”
  • “P là điều kiện cần và đủ để có Q”
  • “Q là điều kiện cần và đủ để có P”

2. Ví dụ minh họa

Xét hai mệnh đề:P: “Tứ giác ABCD là hình vuông”Q: “Tứ giác ABCD là hình chữ nhật có hai đường chéo vuông góc với nhau”

+ Phát biểu \(P \Rightarrow Q\) và mệnh đề đảo:

Mệnh đề \(P \Rightarrow Q\) là: “Nếu tứ giác ABCD là hình vuông thì nó là hình chữ nhật có hai đường chéo vuông góc với nhau”Mệnh đề \(Q \Rightarrow P\) là: “Nếu tứ giác ABCD là hình chữ nhật có hai đường chéo vuông góc với nhau thì nó là hình vuông”

+ Xét tính đúng – sai:

Mệnh đề \(P \Rightarrow Q\) và \(Q \Rightarrow P\) đều đúng.Do đó P và Q là hai mệnh đề tương đương, ta viết \(P \Leftrightarrow Q\)

+ Phát biểu mệnh đề \(P \Leftrightarrow Q\)

“Tứ giác ABCD là hình vuông là điều kiện cần và đủ để nó là hình chữ nhật có hai đường chéo v🦩uông góc với nhau”

“Tứ giác ABCD là hình vuông khi và chỉ khi💮 nó là hình chữ nhật có hai đường chéo vuông góc với nhau”

Quảng cáo

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