Use the quantifiers ‘for all’ (∀) and ‘there exists’ (∃).

Use the diagram to determine which statement best describes the relationship between sets $A$ and $B$.

The diagram shows three mathematical statements involving quantifiers. a) Determine whether statement P is true or false. Justify your answer using algebraic reasoning. (1 mark) b) Determine whether statement Q is true or false by finding a value of \(n\) that satisfies the equation or explaining why no such value exists. (1 mark) c) Use quantifier notation to write the negation of statement R. (1 mark)

Use the diagram to determine which statement is true.

Consider the statement: "For every real number $x$, there exists a real number $y$ such that $x^2 + y = 5$." (a) Use quantifiers $\forall$ and $\exists$ to express this statement symbolically. (1 mark) (b) Determine whether this statement is true or false. Justify your answer. (1 mark) (c) Write the negation of the original statement using quantifiers $\forall$ and $\exists$, and express it in words. (1 mark)

Use quantifier notation to express the statement: "For every positive real number $x$, there exists a positive real number $y$ such that $y^2 = x$."