How do you prove necessity and sufficiency?

We say that the statement A is a necessary and sufficient condition for the statement B when B is true if and only if A is also true. That is, either A and B are both true, or they are both false. Note that if A is necessary and sufficient for B , then B is necessary and sufficient for A . We write A⇔B.

ncl.ac.uk - Necessary and Sufficient Conditions

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Necessity and Sufficiency (Economics)

Necessary and Sufficient Conditions

We use the concept of necessary and sufficient conditions to help us describe relationships between statements.

Necessary Condition

Suppose that $A$ and $B$ are statements. We say that the statement $A$ is necessary for the statement $B$ if $B$ cannot be true unless $A$ is also true. In other words, $B$ requires $A$. However it is possible for $A$ to be true even if $B$ is not true. We write \[A \Leftarrow B\] For example, suppose $A$ is the statement “you sit the exam” and $B$ is the statement “you pass the exam”. You cannot pass the exam without sitting the exam: sitting the exam is a necessary condition for passing the exam. However sitting the exam does not mean that you will necessarily pass the exam.

Sufficient Condition

The statement $A$ is said to be a sufficient condition for the statement $B$ if knowing that $A$ is true guarantees that $B$ is also true. However knowing that $B$ is true does not guarantee that $A$ is true. That is, $B$ needn't be a sufficient condition for $A$. We write \[A \Rightarrow B\] For example, suppose $A$ is the statement “you achieve an overall grade of over $70\%$ in all of the modules that you have studied as part of your economics degree” and $B$ is the statement “you get a first class degree in economics”. Achieving an overall grade of over $70\%$ in all of the modules that you have studied as part of your economics degree means that you will get a first class in economics economics. However, getting a first class degree in economics does not necessarily mean that you achieved a first in all of your economics modules.

Necessary and Sufficient Condition

We say that the statement $A$ is a necessary and sufficient condition for the statement $B$ when $B$ is true if and only if $A$ is also true. That is, either $A$ and $B$ are both true, or they are both false. Note that if $A$ is necessary and sufficient for $B$, then $B$ is necessary and sufficient for $A$. We write \[A \Leftrightarrow B.\] For example, the statement “I am a male sibling” is necessary and sufficient for the truth of the statement “I am a brother”.

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