We are using a grid for combat. In the diagram below, the grey areas are walls.
If A attacked B, would B have any cover from A?
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DMG p. 251 explains the rules for determining whether a creature has cover from an attack when using a grid, under the "Cover" heading:
To determine whether a target has cover against an attack or other effect on a grid, choose a corner of the attacker’s space or the point of origin of an area of effect. Then trace imaginary lines from that corner to every corner of any one square the target occupies. If one or two of those lines are blocked by an obstacle (including another creature), the target has half cover. If three or four of those lines are blocked but the attack can still reach the target (such as when the target is behind an arrow slit), the target has three-quarters cover.
On hexes, use the same procedure as a grid, drawing lines between the corners of the hexagons. The target has half cover if up to three lines are blocked by an obstacle, and three-quarters cover if four or more lines are blocked but the attack can still reach the target.
DMG p. 250 also has a picture demonstrating how this works on a square grid (along with an example for the optional flanking rule):
DMG p. 251 has a corresponding image example for a hex grid.
Given the above guidelines, and assuming the wall corner does not jut out into any adjacent squares at all, A would be able to attack B without B getting any cover from the wall.
The bottom-right corner of A is touching B's space, so a "line" could be drawn from that corner to all 4 corners of B's space; none of the lines are blocked by an obstruction, so B gets no benefit from cover for the attack. (The bottom-left corner of A could equally serve as a point of origin for the lines to the four corners of B, which would still be unobstructed by cover.)
(As always, a DM could feasibly house-rule otherwise - if, for instance, the wall corner doesn't perfectly align to the grid and/or there is an obstruction blocking a direct "line" between the two.)
On a related note, the basic rules do include the following guideline for movement when playing on a grid:
Corners. Diagonal movement can’t cross the corner of a wall, large tree, or other terrain feature that fills its space.
This is entirely up to the DM.
The examples given for half and 3/4 cover lead me to believe that his would probably be half cover, if it provides any cover at all (it wouldn't have in 4th). However, there are no real rules regarding what constitutes half/three fourths cover, so this is left entirely up to DM discretion at this point.
The way I would rule this is that no it does not provide any cover (I plan to import 4e's grid combat rules wholesale, so this would not constitute any cover). However, if the corner between A/B jutted out into either of those squares, suddenly it becomes half cover (2 lines blocked). The way I plan to rule cover is: 1 or 2 lines blocked is half cover, 3 lines blocked 3/4 cover, 4 lines blocked, total cover and you can hide.