Let R and S be two non-void relations on a set A. Which of the following statements is false  R and S are transitive Þ R È S is transitive  R and S are transitive Þ R Ç S is transitive  R and S are symmetric Þ R È S is symmetric R and S are reflexive Þ R Ç S is reflexive

The false statement among the given options is "R and S are transitive Þ R È S is transitive".

Here's why:

1. "R and S are transitive Þ R È S is transitive": This statement is false. The union of two transitive relations is not necessarily transitive. For example, consider two relations R = {(a, b), (b, c)} and S = {(c, a)}. Both R and S are transitive, but their union R È S = {(a, b), (b, c), (c, a)} is not transitive because (a, c) is not in the relation even though (a, b) and (b, c) are.

2. "R and S are transitive Þ R Ç S is transitive": This statement is true. The intersection of two transitive relations is always transitive. If a pair belongs to both R and S, and both R and S are transitive, then the pair will also belong to the transitive closure of R Ç S.

3. "R and S are symmetric Þ R È S is symmetric": This statement is true. The union of two symmetric relations is always symmetric. If a pair (a, b) belongs to R È S, then either it belongs to R or it belongs to S. Since both R and S are symmetric, (b, a) will also belong to R È S.

4. "R and S are reflexive Þ R Ç S is reflexive": This statement is true. The intersection of two reflexive relations is always reflexive. If a relation is reflexive, then every element is related to itself. So, if an element belongs to both R and S, and both R and S are reflexive, then the element is related to itself in R Ç S.