Interpret the predicate Love($x, y, t$) as "$x$ loves $y$ at time $t$". Write the following statements with predicate logic.
(a) Everybody loves somebody at some point. $$(\forall x)(\exists y)(\exists t)\text{Love}(x, y, t)$$
(b) Nobody is in love all the time. $$(\forall x)(\exists y)(\forall t)\lnot\text{Love}(x, y, t)$$
(c) Some people never fall in love. $$(\exists x)(\exists y)(\forall t)\lnot\text{Love}(x, y, t)$$
(d) At some point everyone falls in love with someone. $$(\forall x)(\exists y)(\exists t)\text{Love}(x, y, t)$$
(e) Sometimes two people fall in love with each other forever. $$(\exists x,y)(\forall x)(\forall y)(\forall t)\text{Love}(x, y, t)$$
I am most certain about a through c, d and e were more tricky for mee.
A) correct
B) should be $\forall x \exists t \forall y \neg Loves(x,y,t)$
C) should be $\exists x \forall y \forall t \neg Loves(x,y,t)$
D) move the $\exists t$ to the front
e) get rid of the universal x and universal y