Quantification

We can go from talking about sets to talking about particular elements of sets with the help of quantifiers. In English, a quantifier is a world like most, as in most dogs are happy. Quantifiers are particularly used in first order logic. The most important quantifiers in logic are the Universal (∀) (read “for all”) and the Existential (∃) (read “there exists some”).

Watch the video and then answer the questions below.

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Questions

1. Check your understanding

Given the sets A, B, and C below, are the following expressions true:

A = { 4, 5, 6}
B = { 4, 5 }
C = Ø
1. Existential
ExpressionTrueFalse
1. \( \exists x \in A.x \geq 5 \)
2.\( \exists x \in A.x \in B \)
3. \( \exists x \in C.x \in A \)
4. \( \exists x \in B.x \in A \)

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2. Universal
ExpressionTrueFalse
1. \( \forall x \in A.x \geq 5 \)
2. \( \forall x \in A.x \in B \)
3. \( \forall x \in C.x \in A \)
4. \( \forall x \in B.x \in A \)

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3. Multiple Quantification
ExpressionTrueFalse
1. \( \exists x \in A . \exists y \in B . x = y \)
2. \( \forall x \in A . \exists y \in B . x = y \)
3. \( \exists x \in C . \forall y \in A . x = y \)
4. \( \forall x \in C . \exists y \in A . x = y \)

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4. Quantifiers, Sets and Logic
ExpressionTrueFalse
1. \( ( \neg \exists x . x \in A ) \iff ( \forall x . x \notin A ) \)
2. \( \exists x \in A . \exists y \in B . x = y \)
3. \( \)
4. \( \)

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Summary

In this section we have learned about the operations you can perform on sets.