The domain is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements. You can assume that it is possible for a person to beat himself or herself. a) No one has ever beat Nancy. b) Everyone has been beaten before. c) Everyone has won at least one game. d) No one has beaten both Ingrid and Dominic. e) There are two members who have never been beaten.

A) No one has ever bet Nancy : ¬∈ x B(x, Nancy )

B) everyone has been beaten before : ∈x∀y B(x,y)

C ) Everyone has won at least one game : ∀x∈y B(x,y)

D ) No one has beaten both Ingrid and Dominic : ¬∈x ( B(x,Ingrid) ∧ B(x,Dominic )

E) There are two members who have never been beaten : ∈x∈y (∀z(¬B(z,x) ∧ ¬B(z,y)))

Explanation:

The predicate B ( x, y ) means that person x has beaten person y

No one has ever bet Nancy : ¬∈ x B(x, Nancy ) this expression shows that there is no X that has beaten Nancy

Everyone has been beaten before : ∈x∀y B(x,y) this  expression is because every y has been beaten by someone( x )

Everyone has won at least one game : ∀x∈y B(x,y) this expression is because everybody x beats someone y

No one has beaten both Ingrid and Dominic : ¬∈x ( B(x,Ingrid) ∧ B(x,Dominic ) this expression is because everyone has been beaten and there is no X that has beaten both Ingrid and Dominic

There are two members who have never been beaten : ∈x∈y (∀z(¬B(z,x) ∧ ¬B(z,y))) this expression shows that there exists some members x and y who has never been beaten by Z before

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