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11 June, 18:58

Let the domain of x be the set of geometric figures in the plane, and let Square (x) be "x is a square" and Rect (x) be "x is a rectangle." a. ∃x such that Rect (x) ∧ Square (x). b. ∃x such that Rect (x) ∧ ∼Square (x). c. ∀x, Square (x) → Rect (x).

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  1. 11 June, 19:23
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    a) The statement is also true, because a square is a geometric figure in the plane and a square is also a rectangle.

    b) The statement is also true, because for example a rectangle with length 4 and width 6 is a possible rectangle that is not a square (as the length and the width are unequal).

    c) The statement is true, because squares are a specific type of rectangle and thus all squares are rectangles.

    Step-by-step explanation:

    Negation ~P: not P

    Conjunction p ∧ q: p and q

    Conditional statement p - -> q: if p, then q

    Existential statement ∃xP (x) is true if and only one element x in the domain for which P (x) is true.

    Universal statement ∀xP (x) is true if and only if P (x) is true for all values of x in the domain.

    Domain=set of all geometric figures in the plane

    Square (x) = "x is a square"

    Rect (x) = "x is a rectangle"

    (a)

    ∃x such that Rect (x) ∧ Square (x)

    ∃ mean "there exists"

    In words, the given statement then means that: There exists a geometric figure in the plane that is a rectangle and that is a square.

    The statement is also true, because a square is a geometric figure in the plane and a square is also a rectangle.

    (b)

    ∃x such that Rect (x) ∧ ~ Square (x)

    ∃ mean "there exists"

    In words, the given statement then means that: There exists a geometric figure in the plane that is a rectangle and that is not a square.

    The statement is also true, because for example a rectangle with length 4 and width 6 is a possible rectangle that is not a square (as the length and the width are unequal).

    (c)

    ∀x, Square (x) - > Rect (x)

    ∀ means "for every".

    In words, the given statement means that: All squares are also rectangles.

    The statement is true, because squares are a specific type of rectangle and thus all squares are rectangles.
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